Claims earlier today suggested the Blues had stepped ahead in the race to sign the French midfielder.
Manchester United boss Jose Mourinho has targeted a move for Bakayoko, but opted against an approach in the January transfer window.
According to RTL journalist Bertrand Latour, the Premier League leaders have tabled £32m to Monaco along with £5m in bonuses.
Tiemoue Bakayoko is interested in a move to Chelsea
Bakayoko would certainly add quality to Chelsea's team
Antonio Conte is looking to strengthen his squad at Stamford Bridge ahead of a return to Champions League football.
The Italian wants to be able to compete across all competitions next season, and Bakayoko would certainly add quality to Chelsea's team.
Following reports the Blues had agreed terms with the 22-year-old, it was said their only objective to secure a deal was to pay more than their rivals this summer.
It appears Chelsea are seriously working to see off the likes of United, with Bakayoko apparently interested in the switch.
N'Golo Kante and Tiemoue Bakayoko could form a midfield partnership at Chelsea
Biggest summer transfer odds rated
Wed, March 15, 2017
Get Quotes on Home Insurance
Click to reveal transfer odds for the biggest potential transfer deals this summer. Odds provided by BetStars
Arsenal FC via Getty Images
1 of 5
Alexis Sanchez (Arsenal to PSG) – 13/8
Monaco are yet to respond to the Blues' bid, and if Mourinho is to compete for Bakayoko's signature, he will have to act soon.
Bakayoko recently helped earn Monaco a place in the Champions League quarter-finals after netting a crucial third against Manchester City last week.
And to crown off his achievements in Ligue 1, France boss Didier Deschamps has replaced United's injured Paul Pogba with him in his international squad.