PSG sporting director Leonardo is leading the discussions with Chelsea director Marina Granovskaia as they attempt to pull off a season-long loan with the option to buy.
The Blues may accept a deal with a €3m loan fee but would prefer to either sell the former Monaco star permanently, or include an obligation-to-buy clause.
PSG are the frontrunners ahead of AC Milan, who have found themselves frustrated in their talks with Chelsea over the asking price for the midfielder, who is surplus to requirements at Stamford Bridge.
Thomas Tuchel, the PSG head coach, has approved the pursuit of both Bakayoko and Antonio Rudiger, who is also on the radar of Tottenham, Roma and Barcelona.
"These players belong to other clubs and we have to respect that," Tuchel said at the weekend. "There is no point in talking about these two names. I can only say that the club knows my opinion.
"I am waiting. The transfer window is not over. The situation is not easy, but we still have a little bit of time."
Spurs have become the latest team to enter the race for Rudiger, with the Germany international thought to favour staying in England despite interest from some big clubs overseas.
Jorginho, meanwhile, has also been linked with a move to Parc des Princes but his agent Joao Santos seems to think his client's future remains at Stamford Bridge.
"Jorginho still has a three-year contract with Chelsea, this one plus two more," Santos told 90min.com. "I heard a lot about Arsenal and PSG in the last few days, but the reality is that so far no official offer has arrived for the player.
"The market closes on October 5, but the more time passes the more difficult it becomes to make such an important operation. At the moment, therefore, I can say that it is only rumoured. He is happy at Chelsea, he started the season well and is the vice-captain at the moment."
Chelsea are also looking at offloading Ruben Loftus-Cheek on a season-long loan deal but see his long-term future in west London. West Ham are among the teams interested although his £150,000-a-week wages might be a problem for a number of Premier League sides.
Marcos Alonso's future remains in doubt after he received a dressing down following his half-time substitution away at West Brom. He was blamed for two of the three goals conceded against the Baggies and reportedly irked Frank Lampard by watching the second half from the team bus.
That may see Emerson Palmieri come back into the fold despite Inter having been exploring the possibility of a move, with the full-back having come on as a second-half substitute in Chelsea's penalty shoot-out defeat to Tottenham on Tuesday night.
West Ham, meanwhile, appear to be resistant to any approaches for their star midfielder Declan Rice despite strong interest from Chelsea to add him to their £220m ($275m) summer of spending.