The Fire are believed to be showing significant interest in the 37-year-old striker but could face stiff opposition for the Ivorian's signature from fellow Eastern Conference side Montreal Impact.
Chicago's Chief Operating Officer, Atul Khosla, revealed via Twitter that his club had been in talks with Drogba over the past three weeks.
However, Montreal owner Joey Saputo threw his club's hat in the ring by admitting the Impact were also keen to acquire the services of the former Premier League Golden Boot winner.
"There was a short period in time where Chicago felt they weren't interested in the services of Didier Drogba, for whatever reason," Saputo told TSN Radio this week.
"The league gave other clubs the opportunity to speak to them, so when the league contacted us about Didier, we said 'Absolutely, we're very interested."
Chicago are first in line to sign Drogba, given the veteran was on the club's 'Discovery list'. MLS rules stipulate a Discovery process in which each MLS club submits a list of seven unsigned players to the MLS league office, thus establishing priority on a first-come, first-serve basis.