The Blues have been heavily linked with a move for the player in recent weeks and now appear to have made their interest in him official.
Coach Jose Mourinho is an admirer of the 27-year-old and Malouda has already stated his desire to leave Stade Gerland.
A deal to bring the player to Stamford Bridge now looks increasingly likely, although Chelsea's offer is still below Lyon's valuation.
Meanwhile, Portsmouth technical director Avram Grant is set to join the club on July 1, although Chelsea refused to comment on his impending arrival.
The former Israel manager is friends with Chelsea's billionaire owner Roman Abramovich and speculation about Grant's arrival sparked a row between Mourinho and the club's owner back in January.
But the differences have been resolved and Grant is expected to be given a wide-ranging role as director of football.
Negotiations with Chelsea are alleged to have started, although no contract has been signed as yet.
A Chelsea spokesman said: "Avram Grant is an employee of another football club and, as such, we will not comment."
Portsmouth manager Harry Redknapp has made it clear he would not stand in Grant's way and the Israeli is expected to agree a contract with the Blues by the end of the month.
Elsewhere, Newcastle boss Sam Allardyce wants Chelsea's Cameroon international Geremi.
Geremi had been linked with a move to north-east rivals Middlesbrough, where he once spent a season on loan, but Allardyce is hopeful of concluding a deal for the 28-year-old.
The Cameroon defender is approaching the end of his contract at Chelsea and will be available on a free transfer after failing to hold down a regular first team place at Stamford Bridge.
Chelsea's reserve team goalkeeper Yves Makabu-Ma Kalambay is to leave the club and join Hibernian on July 1.
The 21-year-old Belgian was signed from PSV Eindhoven in 2003 but did not manage to break into the first team squad. He played 41 reserve team games - including 11 last season - and 30 in the youth team.
Hibs boss John Collins said: "He has benefited from world-class coaching at Chelsea under Silvino Louro but such is the quality of goalkeepers at Chelsea there was little opportunity for first-team football and at this stage in his career that is what Yves is seeking."
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