City are understood to have made the bid last week and while the club rejected the Premier League winners’ offer, Agger is reported to have told new Reds manager Brendan Rodgers that he wants to leave Anfield.
Aware of City’s interest, Agger is likely to wait before making a decision about where he will be playing next season.
City boss Roberto Mancini is keen to add defensive reinforcements to his squad and has identified Agger being the ideal candidate to provide competition for Vincent Kompany and Joleon Lescott.
Kolo Toure is the player likely to be sacrificed by Mancini, with the Ivory Coast international likely to leave the Etihad Stadium this summer.
The former Arsenal defender joined City for 15-million pounds but is now being targeted by Turkish side Bursaspor and Saudi Arabia’s Al Hilal.
Barcelona are also said to be interested in Toure, with new boss Tito Vilanova keen on adding a versatile and experienced player to his side.
City have so far been unable to find a buyer for Toure willing to match his 120,000-pounds-a-week wages.
Chelsea are looking to complete more signings this summer, according to club officials, with the Champions League winners on the cusp of completing three transfers.
Despite spending 65-million pounds bringing Eden Hazard, Oscar and Marko Marin to Stamford Bridge, the West London side have not yet put away their cheque book.
Chelsea chief executive Ron Gourlay has confirmed that the club plans to continue its summer spending spree.
“It was clear there were areas where we had to strengthen if the right players became available,” Gourley said.
“We've done that and are still looking at one or two positions.”
The club is currently in negotiations to sign Edinson Cavani from Napoli in a deal worth 32-million pounds, while also discussing the transfers of Wigan’s Victor Moses and Marseille’s Cesar Azpilicieta.
Manchester United are considering a move for Chelsea’s Ashley Cole, with United boss Sir Alex Ferguson seemingly ending the club’s pursuit of Everton’s Leighton Baines.
Ferguson is determined to sign a fullback this summer and had made Baines his first choice but with Everton unwilling to budge on their 15-million-pound asking price the deal is said to be off.
British daily Metro reports that United are now looking at alternatives to Baines with Cole seen as a more realistic prospect.
Cole has just 12 months remaining on his deal and despite excellent performances for the club, he has not been offered an extension to his contract.
No talks have been scheduled between the clubs but it is likely that Chelsea will soon decide whether to sell the 31-year-old or let him leave for nothing next year.
QPR are set to bid for Arsenal’s Andrey Arshavin and Tottenham’s Jermain Defoe, with Hoops boss Mark Hughes hoping that the lure of first team football will be enough to encourage the two to move to Loftus Road.
Both players struggled to command first team places with their respective North London clubs last season, and caughtoffside.com reports that QPR are in the frame to land the two forwards.
Arshavin is known to want to remain in London should, as expected, he leave the Emirates Stadium this summer, although Fulham are expected to rival QPR for the Russian’s services.
England striker Defoe rarely made Spurs’ starting lineup last season, with former boss Harry Redknapp preferring to use the player as an impact substitute.
Defoe is a proven goal scorer in the Premier League, claiming 17 goals in all competitions last season despite most of his appearances coming from the bench.