The French midfielder, 26, has turned down a new deal at Fulham - believed to be worth £40k a week - which has forced the Londoners to listen to offers with Malbranque having just 1 year left on his contract.
Liverpool tried to sign Malbranque 2 years ago and the Mirror reports that they have made an enquiry again, with both they and City having begun negotiations.
City are listening to offers for Joey Barton, but Newcastle today ruled out a move for the combative midfielder almost as quickly as they had been linked with him.
Aston Villa boss David O'Leary also insisted he had no interest in signing Danny Mills after being linked with the veteran City right-back.