French newspaper L'Equipe claims the FA Cup winners have increased their offer from 18.5million euros (£12.5million) to 20million euros (£13.5million) to secure the signature of the 27-year-old.
It is thought the France international will sign a four-year deal worth 3.5million euros-a-year (£2.37million) on Sunday or Monday ahead of Chelsea's pre-season tour to the USA.
L'Equipe reports Lyon were hoping Real Madrid might spark a bidding war for the player, but he is intent on moving to the Premier League and only contractual details between the French club and Malouda are holding up the move.
Chelsea chief executive Peter Kenyon had previously admitted he hoped the deal would be concluded in time for Malouda to link-up with the rest of the squad during the Blues' trip to the States.
"We've been in dialogue with Lyon and we're looking to conclude that," said Kenyon. "It would be nice to conclude that before we go to Los Angeles.
"I'm very optimistic he will be joining us at some point out there."
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