The Uruguay star has made numerous public pronouncements since the end of the season, suggesting a move to Real Madrid interests him and claiming his mistreatment at the hands of the British media has driven him to look elsewhere.
Carragher, though, believes Suarez may yet stay at Anfield.
“I think he might stay. He signed a contract last summer and he has a few years left and hopefully we can keep him, He is very important. He was probably the best player in the Premier League last season and you want to be keeping your best players, which is what he is.”
While the Suarez transfer saga seems likely to run and run, Bayern Munich winger Franck Ribery has delivered on his promise by putting pen to paper on a new four-year contract at the European champions.
“I’m really pleased,” he said. “I always said how great it feels to be at FC Bayern. I have a good feeling for me and my family and I hope things will carry on the way they were in the past year.”
Elsewhere, Stoke have completed the signing of Netherlands defender Erik Pieters from PSV Eindhoven. The 24-year-old has signed a four-year contract and will cost Stoke a fee of €3.6 m.
Pieters becomes Stoke manager Mark Hughes’ first new signing since he succeeded Tony Pulis in May.
Newly-promoted Hull were also active yesterday as winger Ahmed Elmohamady completed his move from Sunderland. The 25-year-old Egypt international, who enjoyed a successful loan spell at the Tigers last season, has signed a three-year deal after the two clubs agreed an undisclosed fee.
Marko Marin is also on the move after Chelsea announced the German international will spend next season on loan at Sevilla.
Meanwhile, Derby manager Nigel Clough has admitted he would be interested in signing Richard Dunne but doubts the Republic of Ireland defender would be willing to drop down to the Championship. Dunne is available on a free transfer after being released by Aston Villa after failing to play for the Premier League club last season due to injury.
He did return to action for the Republic of Ireland against Georgia at the start of the month and is now looking for a new club ahead of next season. Clough feels the 33-year-old is keen to stay in the top flight, but acknowledged the former Manchester City defender is the type of player he would be interested in signing.
Elsewhere, Ian Harte, a former international team-mate of Dunne, has joined Championship newcomers Bournemouth on a one-year contract.
Harte was released by the Royals in May following their relegation from the Premier League after making 99 appearances in three years at the club.