The 21-year-old has played an integral role in Monaco's superb 2016-17 season which sees them two points off the summit of Ligue 1, having comfortably progressed into the knockout stages of the Champions League.
According to the Daily Telegraph, Bakayoko is on a shortlist of four players Jose Mourinho has identifed as the club's next transfer targets over the next two windows alongside Benfica duo Victor Lindelof and Nelson Semedo alongside Atletico Madrid forward Antoine Griezmann.
But Chelsea are also monitoring the France Under-21 international with the Guardian reporting Antonio Conte wants the midfielder and his Monaco teammate Djibril Sidibé as he continues his revolution at Stamford Bridge.
Both Premier League sides will have to wait until the summer, however, with Bakayoko insistent he will not abandon Monaco half-way through the season.
"I have had a good first six months, but the season is not over. I won't leave Monaco this winter, that's a certainty. I don't want to leave," he told Eurosport.
"Of course the Premier League makes you dream, but I have no reason to leave this winter. Right now, I have two ambitions: finish champions of France and go as far as possible in the Champions League."
With Bakayoko seemingly unavailable at this point, United could be in for a quiet January window. The club have reportedly cooled their winter interest in Lindeolf, with Mourinho satisfied with his current options of Phil Jones, Marcos Rojo, Eric Bailly and Chris Smalling.
At Chelsea, Conte is keen to boost his squad to ensure their Premier League title charge does not fade. "We must be very pleased for the achievement we have reached now, but it is logical that I look with my club for a solution to arrive in a good position at the end of the season," Conte told Chelsea's official website.
"I don't know what will happen in January but in this situation it is important to talk with my club – we have a fantastic relationship – but I have my idea and I prefer to discuss this with the club."