The BBC understands Fulham hope to have a successor in place in time for their pre-season training camp in mid-July.
Club captain Gerrard believes Hodgson's extensive experience in Europe make him the "right man for Liverpool".
"I think it's been worth the wait and I'm sure he's just keen now to get on with it and start to quickly put in place his plans for the new season," said the 30-year-old.
One of Hodgson's first priorities is securing the futures of Gerrard, star striker Fernando Torres and midfielder Javier Mascherano.
All three players, who have been linked with big-money moves away from Anfield, are vital to Liverpool's hopes of reviving their Champions League ambitions having missed out on a place in next season's competition after finishing seventh in the Premier League.
And Liverpool chairman Martin Broughton assured Hodgson he would not have to offload players to service the club's huge debts of about £350m.
I will be trying to persuade them that their future is with the club they love and have always played for
Hodgson outlines what he will say to Gerrard and Carragher
"Players are not for sale," Broughton told the news conference. "We don't need the money and we don't have to sell them because of the debt."
Hodgson, who is not allowed to sign Fulham players as part of his severance deal with the London club, added: "I don't think there's anybody here at the club who would welcome losing them.
"Certainly the fans wouldn't welcome losing them and as a club I'm sure will do everything we can to make certain they stay with us.
"I'll be trying to persuade them that this is the place to be. I'll ask them to give me a chance to work with you and give us a chance to improve on last season.
"They're all disappointed, I'm sure, after last season's results and I'll be doing my level best. It would be foolish to give guarantees that it's going to happen because I can't speak for the players themselves, but I'll be doing my best to keep them."
Liverpool legend Kenny Dalglish, who is now an ambassador at the club and was linked with the manager's job, insisted he welcomed Hodgson's appointment.
"I look very favourably on it," he told the Daily Mail. "I have known Roy a long time and have a great respect for him. We've not had detailed conversations yet but my interests are in Liverpool."
Dalglish will now be given a new role, as the main link between the academy at Kirkby and the senior squad at Melwood.
HODGSON'S MANAGERIAL CAREER
1976-1980 Halmstad (Swe)
1982 Bristol City
1983-1985 Örebro (Swe)
1985-1990 Malmö FF (Swe)
1990-1992 Neuchâtel (Swi)
1995-1997 Inter Milan (Ita)
1997-1998 Blackburn Rovers
1999 Inter Milan (Ita)
1999-2000 Grasshopper (Swi)
2000-2001 Copenhagen (Den)
2001 Udinese (Ita)
2002-2004 United Arab Emirates
2004-2005 Viking (Nor)
2010 - Liverpool
Hodgson was due to meet club stalwarts Gerrard and Carragher after the news conference, and revealed he would implore them to stay at the club.
"My talk with them will be about how much I want to work with them, how much I want them to stay at the club, how much the club and fans want them to stay and that by staying at the club we can achieve as much success as they could find by leaving," Hodgson said.
"That is all I can say to them. Then of course it is a case of making certain with your actions on the training ground and match days they see you are not just talking but are practising what you preach.
"I will be trying to persuade them that their future is with the club they love and have always played for.
"They are more than just Liverpool football club players, they are the heartbeat and very much the soul of this team.
"I hope they won't be tempted, on the back of one disappointing year, to jump ship.
"I personally can't nail their feet to the floor. I'll try to convince them that if they stay here and have a good season then it's a good club to play for."
Carragher said he welcomed the appointment of Hodgson after a turbulent period in the club's history.
"I'm really looking forward to the opportunity of working with Roy Hodgson as he's got such a great reputation in the game and is one of the most experienced managers in world football," said the England centre-back.
"After this period of uncertainty, his appointment has given us all a lift."
Liverpool moved quickly after Hodgson was linked with the England managerial job after the team's World Cup failure.
Hodgson has extensive domestic and international managerial experience across Europe, including spells in charge of Blackburn Rovers and Switzerland, who he guided to the last 16 of the 1994 World Cup.
His departure is a bitter blow for Fulham, with whom Hodgson signed a 12-month rolling contract in December last year.
Hodgson arrived at Craven Cottage in December 2007 after the sacking of Lawrie Sanchez, guiding the club clear of relegation in his first season back in the Premier League.
He then forged a competitive unit with astute signings like Norway defender Brede Hangeland, Australia goalkeeper Mark Schwarzer and Hungary midfielder Zoltan Gera.
His exploits with Fulham last season, which included wins over Juventus and Wolfsburg en route to a 2-1 Europa Cup final defeat by Atletico Madrid in Hamburg, earned him the League Managers' Association manager of the year award in May.
"The club would like to wish Roy the very best for the future and thanks him for all that was achieved during his tenure," read a club statement.
"(Assistant manager) Ray Lewington will take temporary charge of team matters, until a successor is announced in due course."
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