The Premier League leaders are understood to have rejected an approach from the Anfield club, who are thought to be looking to offload the striker during the January window
Liverpool have offered Andy Carroll to Manchester City in a straight swap deal for wantaway striker Carlos Tevez, the Guardian reports.
The paper claims the Premier League leaders' football administrator, Brian Marwood, rejected the approach outright from the Merseyside club, which was made on Thursday.
Carroll, who was a £35 million deadline day signing in January last year, has scored just six goals in all competitions for the club since his move to Anfield, but did start in the FA Cup fourth round victory over Manchester United on Saturday afternoon.
Meanwhile, Tevez remains with Roberto Mancini's side despite his desire to leave and the club's willingness to offload the Argentina international.
Serie A club Inter have withdrawn from negotiations to sign the striker, as have French side Paris Saint-Germain.
AC Milan remain an outside option with just three days remaining of the January transfer window, though their signing of Catania's Maxi Lopez makes a winter exit for Tevez now highly unlikely, with City chairman Khaldoon Al Mubarak insisting the club will not budge from their valuation of the player.
Carroll was on the bench and a late introduction to the League Cup semi-final second leg between the two sides on Wednesday evening.
Kenny Dalglish has been a staunch defender of the striker publicly, but another recent report, which was vehemently denied, suggested Carroll had been offered back to Newcastle for a fraction of the £35m his old club received from the transfer just 12 months ago.