With the Democratic National Convention set to begin on 25 August, the party's presumptive nominee for the presidential election, Barack Obama, is expected to pick a running mate any day now.
Whoever he chooses will become vice-president if he wins the election in November.
The BBC News website takes a look at some of the people who may be on his shortlist.
Senator Joe Biden ran against Mr Obama and Ms Clinton for the presidential nomination but dropped out after failing to gain enough support.
Status: Delaware senator, chair of Senate Foreign Relations Committee
Advantage: Foreign policy experience, strong debater
Disadvantage: A Washington insider, could undermine message of "change"
He is a foreign-policy heavyweight who has advocated a federal solution to the conflict in Iraq, envisaging a loose division of the country along ethnic lines.
A powerful orator, he ran for president briefly in 1988 but withdrew after he admitted to plagiarising a speech by Neil Kinnock, who was the leader of Britain's Labour party at the time.
He is also prone to the occasional gaffe. In 2007, he described Barack Obama as "the first mainstream African-American who is articulate and bright and clean and a nice-looking guy".
Mr Biden has been a senator, representing the small north-eastern state of Delaware, for some 35 years - longer than John McCain.
His foreign policy expertise could help Mr Obama defend himself against Republican attacks that he is not ready for the role of commander in chief.
Republicans may well argue that the choice of a veteran running mate only serves to highlight Mr Obama's own inexperience and undermines his promise of change.
Jim Webb's victory in the 2006 Virginia Senate race was one of the key moments of the Democrats' congressional takeover that year.
Since then, Mr Webb - a Vietnam veteran who served as Ronald Reagan's Navy Secretary in the 1980s - has built up a name for himself in the Senate as a man to watch.
Status: Virginia senator
Advantage: Could help deliver swing state of Virginia
Disadvantage: Has ruled himself out
Like Mr Obama, he opposed the war in Iraq, and with his military Scots-Irish background, picking him might help Mr Obama with white-working class voters.
But, more importantly, it would give him a boost in Virginia, a state which has been trending Democratic in recent years, and that - with his strong base of support among African-Americans - Mr Obama has a chance of winning in November.
However, Mr Webb has ruled himself out of the running, saying: "Under no circumstances will I be a candidate for Vice President".
So if Mr Obama really wants to pick Mr Webb he has a lot of persuading to do.
If Mr Obama wants to choose someone who - like Senator Jim Webb - could help him win Virginia, then Governor Tim Kaine could well fit the bill.
Status: Virginia governor
Advantage: Could help to deliver Virginia, a key swing state
Disadvantage: Relatively inexperienced
And picking Mr Kaine, unlike Senator Webb, would not leave the Democrats fighting to replace a senator.
Mr Kaine was a early backer of Mr Obama, and his loyalty might count for something.
But although popular, he has only been the state's governor since 2006, and Mr Obama may decide that the ticket needs to have a bit more experience on it.
Indiana Senator Evan Bayh was one of Hillary Clinton's most high-profile backers during the primaries, so picking him as a running mate might help Barack Obama build bridges with Clinton-supporting Democrats.
Status: Indiana senator
Advantage: Swing-state senator, who could help win back Clinton supporters
Disadvantage: Critics say he lacks charisma
Having Mr Bayh on the ticket could also make his home state of Indiana more competitive for the Democrats.
But when he was considering a presidential run himself last year, some observers suggested that Mr Bayh did not quite have the charisma needed to reach the top of the US political heap.
However, if Mr Bayh is able to overcome this reputation, then he could well find himself near the top of Mr Obama's shortlist.
Many observers have suggested that Mr Obama may well be keen to put a woman on his ticket - just not his erstwhile rival.
Status: Kansas governor
Advantage: Executive experience
Disadvantage: Unlikely to deliver Kansas
If he does pass over Mrs Clinton, one possible female running mate could be Kansas Governor Kathleen Sebelius.
Mr Obama has a warm personal relationship with Ms Sebelius, and she is well-rated for her time as Kansas governor.
Having someone with executive experience could help the presidential candidate, who has only ever served in legislative bodies.
But Ms Sebelius is unlikely to be able to deliver any states for Mr Obama - even her home state is likely to be out of reach for the Democrats.
Furthermore, some observers suggest that picking a woman other than Mrs Clinton would serve only to alienate her - and her supporters - even further.
Bill Richardson's endorsement of Barack Obama after his own departure from the presidential race lost him plenty of friends in the Clinton camp.
Status: New Mexico governor
Advantage: Executive experience, foreign policy experience, popular with Hispanics
Disadvantage: Reputation as a loose cannon
But it will have done his chances of becoming Mr Obama's running mate no harm at all.
He has been a popular governor in the swing state of New Mexico, and he would be able to help the Obama campaign throughout the south-west, a part of the country where Mr Obama is hoping to make gains.
In this Mr Richardson will be helped by his Hispanic heritage, but this ethnic background might be an electoral hindrance in other parts of the country.
However Mr Obama might choose to emphasise his own historic candidacy - as the first African-American nominee - by choosing a running-mate who is also non-white.
But observers say Mr Richardson has a reputation for being something of a loose cannon. The Obama team may have to be prepared for one or two gaffes if they decide to put his name on the ballot.
Ed Rendell, the plain-spoken governor of Pennsylvania, was one of Hillary Clinton's most prominent supporters during the primaries, and his advocacy - and campaign machine - helped her secure victory in the state.
Status: Pennsylvania governor
Advantage: Appeal to white working class and Clinton loyalists, and would help hold Pennsylvania
Disadvantage: Too close to Clinton?
Picking Mr Rendell could allow Barack Obama to kill three birds with one stone - broaden his appeal to white working-class voters, improve his chances of keeping Pennsylvania's votes in Democratic hands, and mollify Clinton loyalists.
He would also bring some executive experience to the ticket.
But his close connection to Mrs Clinton could harm his chances. Mr Obama might feel more inclined to reward someone who backed him from the very beginning of the campaign.
A well-respected former senator from Georgia who served as chairman of the Armed Services Committee, Sam Nunn has been suggested as a running mate who might reassure voters concerned that Barack Obama does not have enough experience, especially in military matters.
Status: Former Georgia senator
Advantage: Experience, foreign and defence policy credentials
Disadvantage: The Obama ticket would look less young and new
Since leaving the Senate, Mr Nunn has spearheaded attempts to restrict global nuclear proliferation, a cause with which Mr Obama has also been heavily involved, and he was an early backer - and unpaid adviser - to the Obama campaign.
But - although he is two years younger than John McCain - his age could be a drawback for a campaign which wants to project itself as youthful and new.
Ted Strickland, like Ed Rendell, is a popular, Clinton-backing governor of a rust-belt swing-state.
The difference is that while Democrats won Mr Rendell's state of Pennsylvania last time round, and are favoured to win it again, they narrowly lost Mr Strickland's home-state of Ohio in 2000 and 2004.
Status: Ohio governor
Advantage: Help to capture a crucial swing state, Ohio
Disadvantage: Has ruled himself out
Having Ohio's governor on the ticket might just make the difference for Mr Obama between winning and losing the state.
But Mr Obama has said he will be looking for a running mate with whom he has a personal connection, and there is no indication that the two men have any such bond.
Plus, Mr Strickland has ruled himself out, saying, "if drafted, I will not run, nominated, I will not accept, and if elected, I will not serve".
So if he does end up on the ticket, he will have a lot of back-pedalling to do.
Having come so close to the nomination herself, Hillary Clinton has been talked of as the most obvious running mate for Mr Obama.
Status: New York senator, former first lady
Advantage: Unites party
Disadvantage: Blunts "change" message
Adding her to the ticket would heal the divisions within the party, the argument goes, and help Mr Obama to hold on to Mrs Clinton's loyal supporters, some of whom might otherwise be tempted to defect from the Democrats.
This is an argument pushed publicly by many of Mrs Clinton's backers, although she has distanced herself from their calls.
Picking Mrs Clinton might, however, make Mr Obama look as though he was reneging on his central campaign message - to bring change to Washington.
He might also be wary of putting such a high-profile person - with an equally high-profile husband - on his ticket, who could overshadow him during the campaign and, if he wins, throughout his presidency.