The 37-year-old failed to turn up to Man United's pre-season training, citing family reasons.
Last month Todd Boehly had a meeting with Ronaldo's agent Jorge Mendes and a possibility of bringing the player to Chelsea was one of the topics of discussion.
Since then Boehly and Mendes have been in continued conversation about the move. Boehly's fellow director Behdad Eghbali is also intrigued by the possibility of bringing a player of Ronaldo's stature to Chelsea.
The final decision would fall upon manager Thomas Tuchel however, as Boehly will not okay any move without his blessing.