The News of the World claims that the Argentine forward would be in favour of a move to west London after handing in a written transfer request at Eastlands.
The former Manchester United striker's request has since been rejected, and the player himself is thought to have assured manager Roberto Mancini that he will remain professional and put in his best efforts for the greater good of the club.
However, there appear to be doubts as to whether the 26-year-old will really fulfil his contract at City - which could encourage Chelsea to formalise an offer.
The Roman Abramovich-owned club are also reported to be trailing Wolfsburg's Dzeko.
The Bosnian has been linked with moves to several elite club in Europe following his exploits in the Bundesliga, which included a fantastic run with the team as they won their first ever league title in 2008-09.
It is thought that the towering hitman has a buy-out clause in his contract that stands at £34 million. But the faltering form shown by current Chelsea strike pair Didier Drogba and Nicolas Anelka during the club's recent rough patch in the league is reportedly prompting the Blues to consider drastic remedial measures.
It is claimed that Chelsea will move for both Tevez and Dzeko in January if either club should show any willingness to trade.
The Blues' forward has 18 months left on his current deal at Stamford Bridge, and will be 34 by the time that contract expires.
The News of the World claims that the upper echelon at the club are reluctant to bow to the forward's desire to play for two more years, owing to his diminished form in recent weeks, though the effects of a bout of malaria may well have contributed to that.
There have been suggestions that the Ivory Coast international is past his prime, with some quarters citing the fact that the hitman was benched for last week's crucial encounter against Tottenham Hotspur.
To his credit, however, the former Olympique Marseille man came on to score the leveller. And he had the opportunity to grab the winner, too, with a spot-kick, though the 32-year-old was unable to convert the penalty and the game ended all square at one-apiece.
Drogba is currently 16th in the Castrol Rankings, down from ninth the previous month.
Meanwhile, there are also reports that the Chelsea squad may have lost some of its unity following the sacking of popular assistant manager Ray Wilkins. His dismissal has coincided with a slump in form that has seen the team slip from first to fourth and fail to win any of their last five Premier League fixtures.
the problem with chelsea is not the strikers!
if they cant see this, they are blind.
not a bad idea to add on some quality up front. afterall drogba is getting old, kalou and malouda are wingers, & dont really do as well as a good striker does up front.