With the future of current No.1 Kepa Arrizabalaga in doubt, the Blues have identified Mendy as an ideal option to provide competition for the Spaniard.
The Sun reported earlier in the week that Rennes turned down Chelsea's offer of €17m (£15m/$20m) for Mendy, and Frank Lampard's side have now returned with an improved bid.
The Blues made Kepa the most expensive goalkeeper ever in 2018, paying £72 million ($93m) to bring him to Stamford Bridge from Athletic. However, the 25-year-old has failed to live up to expectations in London.
Lampard dropped Kepa late in the 2019-20 season after a string of inconsistent performances, opting for veteran Willy Caballero instead.
But Kepa's wages could prove problematic for the Liga sides, meaning Chelsea could have three goalkeepers in their first-team squad should Mendy arrive.
Rennes are determined to hang on to the 28-year-old, however, after the Senegalese international helped his team to their first ever Champions League spot with a third-place finish in Ligue 1 last term.
Mendy only arrived at Rennes last summer following several solid campaigns with Stade de Reims, and has a contract through 2023.
Rennes have yet to respond to Chelsea's bid, as Lampard continues to spend big in the hope of building a contender for the 2020-21 Premier League title.
The Blues have already brought in a host of new faces ahead of the start of the season including Ben Chilwell, Timo Werner, Hakim Ziyech, Malang Sarr and Thiago Silva.
Their most expensive signing appears yet to come as well, as they continue to close in on a move for Bayer Leverkusen star Kai Havertz.
The 21-year-old would cost Chelsea £90m ($118m), with £72m ($95m) coming up front and a further £18m ($24m) in instalments.