City boss Mark Hughes made a firm inquiry about the Ivory Coast striker in January and, although he was knocked back, the Londoners admitted a summer move is a possibility.
Now, City wants to push through a three-year deal, worth around 150,000 pound a week, as soon as the window reopens.
Hughes is desperate to recruit a powerful front man after cooling his interest in Blackburn's Roque Santa Cruz, The Sun reports.
Drogba's Stamford Bridge [ Images ] future was up in the air even before last week's foul-mouthed television rant following their UEFA [ Images ] Champions League [ Images ] knockout by Barcelona [ Images ].
There is a split within Chelsea over whether to keep him or not but City's offer for a 31-year-old, who has been plagued by injury problems, would be hard to ignore.
Meanwhile, Hughes insists he has no fears about his future at City. The Eastlands boss said: "People talking about me and my job is just water off a duck's back.
"But from within the club I feel I'm getting some credit. I have two years left on my contract at the moment. My view is I'll be here next year," he added.