The audacious move was made after the Carling Cup semi-final second leg between the two clubs at Anfield, with the Premier League leaders rejecting the proposal.
Carroll, 23, has struggled to make an impact since Liverpool made him the most expensive British footballer of all time last summer when signing him for £35 million from Newcastle.
Roberto Mancini has previously expressed his admiration for the England striker, when he played for the Magpies, which prompted the Merseysiders to ask if City would consider a swap deal.
But City have maintained throughout the long-running Tevez transfer saga that they will only accept a cash bid for their wantaway striker, and that stance was reiterated to the Reds.
"It didn't go any further than an enquiry," ESPNsoccernet sources said on Sunday.
Tevez's advisors have indicated that he plans to return to City's training ground on Wednesday, once the transfer window closes, though ESPNsoccernet understands City have yet to be informed precisely when the Argentina international plans to return.