The defending Premier League champions have been attempting to negotiate a transfer for Drinkwater for several weeks now, with Bakayoko's old knee injury and a raft of departures leaving manager Antonio Conte short of options in the middle of the park.
SFR Sport claimed over the weekend that Chelsea and Leicester had settled on a deal for the 27-year-old worth in excess of €30m (£27.8m, $36m), with the latter said to be targeting Paris Saint-Germain midfielder Grzegorz Krychowiak as a potential replacement.
The addition of another high-profile midfielder could be viewed as a potential challenge to Bakayoko, although the French international evidently harbours no concerns over the increased competition.
"Competition only makes you better. I am not scared of anyone," he told SFR Sport, as translated by Sky. "I think that there are not enough of us in midfield, especially with all the competitions we will be playing in. So we need another midfielder to reinforce the team, so it is a very good thing that he could be coming."
Drinkwater's arrival at Chelsea would see him reunite with N'Golo Kante and look to resume the effective midfield partnership that was so integral to Leicester's fairytale top-flight title success in 2015-16.
The Manchester United academy graduate has made 218 club appearances and earned three senior caps for England since swapping Old Trafford for the King Power Stadium in 2012. He is yet to feature for Leicester this season due to a thigh strain, although did return to the training pitch last week.
Chelsea look set for a very busy end to the current transfer window after a frustrating period in which a lack of activity sparked fresh rumours that an increasingly frustrated Conte could exit after just one year. In addition to continuing their pursuit of Drinkwater, the club are also said to have agreed a fee of around £35m for wantaway Arsenal midfielder Alex Oxlade-Chamberlain and hope to sign Swansea City striker Fernando Llorente.
Injured Everton playmaker Ross Barkley, Southampton duo Virgil van Dijk and Cedric Soares and Inter Milan's Antonio Candreva have also been linked with prospective moves to Stamford Bridge, while AWOL striker Diego Costa's long-awaited return to Atletico Madrid appears to be edging ever closer. Loic Remy and Kenedy are among the other players that could depart before the end of the week.
Chelsea have made four new additions so far this summer, with Bakayoko joined in west London by Willy Caballero, Antonio Rudiger and Alvaro Morata.