Chelsea owner Todd Boehly reportedly met with Cristiano Ronaldo's agent Jorge Mendes last week.
Boehly is personally taking charge of Chelsea's transfer business this summer after recently becoming the club's new chairman and interim sporting director.
According to The Athletic, Boehly and Mendes met in Portugal and discussed, among other things, the idea of Ronaldo joining Chelsea.
Ronaldo, 37, has 12 months left on his current contract at United and has not said publicly that he wishes to leave.
United want to keep Ronaldo, while the player himself has a strong affinity with the Old Trafford club.
But Ronaldo is said to be underwhelmed by United's lack of transfer activity so far this summer, especially as last season was the club's worst in the Premier League era.
The Athletic's report admits that it is not yet clear whether Chelsea will make an offer for Ronaldo, who has also been linked with Bayern Munich.
But a new central striker is likely to be on Chelsea's wish-list as Romelu Lukaku is set to rejoin Inter Milan on loan.
Mendes represents many of Portugal's top players. His other clients include Ruben Dias, Bernardo Silva, Joao Cancelo, Diogo Jota, Goncalo Guedes, Ruben Neves, Andre Silva, Pedro Neto and Renato Sanches.