Most of those killed in the Lockerbie bombing were Americans
The US and Libya have agreed to work together to resolve compensation claims from the Lockerbie bombing and other 1980s attacks blamed on Libyan agents.
Libya has already paid out $8m (£4m) to each Lockerbie victim's family but has not made final payments of $2m amid a dispute over America's obligations.
A US court ruling that Libya should pay billions of dollars to Americans killed by another bomb incensed the Libyans.
They are hoping for an all-in-one deal to cover that and the other attacks.
They are also said to be wary of a new US law allowing victims of terrorism to seize US-held assets of states held responsible.
Libya reportedly complained that it was being punished rather than rewarded for its decision in 2003 to stop working on weapons of mass destruction.
That decision led to the restoration of US diplomatic ties with Libya, which was removed from America's list of countries sponsoring terrorism.
But Libya feels the US is not living up to its obligations and wants more political and economic benefits from its new ties with Washington, the BBC's Kim Ghattas reports from Washington.
'Not a penny less'
News of the agreements on negotiations came in a joint statement issued by the US state department:
"Representatives of the United States and Libya met in London May 28-29 to begin negotiations on a claims settlement agreement.
"Both parties affirmed their desire to work together to resolve all outstanding claims in good faith and expeditiously through the establishment of a fair compensation mechanism."
A US state department official, speaking anonymously to the Associated Press ahead of the joint statement, said both nations wanted "legal peace" and a clean slate.
The proposed deal would address the 1988 bombing of Pan Am plane over Lockerbie and at least five other attacks, including the 1989 bombing of a French UTA airliner in which seven Americans were killed.
In January, a US court ordered Libya and six Libyan officials to pay more than $6bn to relatives of the seven US victims aboard the UTA plane.
Libya had already agreed to pay $1m compensation to the relatives of each of the 170 people on board the flight though it denied any link to the bombing.
Abdelbaset Ali Mohmed al-Megrahi, a Libyan, was convicted in 2001 of blowing up Pan Am Flight 103.
Reacting to Friday's joint announcement, lawyers for 130 of the Lockerbie victims said they welcomed the proposed talks but insisted the outstanding $2m sums should be paid.
"I don't think any of the clients would accept a penny under $2m," said Jim Kreindler, head of the law firm involved.