ESPN reported on July 4 that the 37-year-old was open to moving to Stamford Bridge after expressing his desire to leave United amid frustration over the club's lack of transfer activity.
Ronaldo's agent, Jorge Mendes, met with Chelsea chairman Todd Boehly, who has since discussed the proposal with head coach Thomas Tuchel.
Sources have confirmed to ESPN that while Tuchel has huge admiration for Ronaldo, he would prefer to pursue other options to strengthen their attack.
Ronaldo, meanwhile, is set to turn down a mega-money offer from Saudi Arabia, sources have told ESPN FC. An unnamed Saudi Arabian club has made a €30m bid to United to sign Ronaldo while giving the Portugal captain the chance to become the world's highest-paid player by offering him €275m in wages for the next two seasons.
Sources told ESPN FC that the former Real Madrid star wants to play Champions League football next season and that competing for major trophies in a competitive league is essential to him at this stage of his career.
Chelsea confirmed the £50 million capture of Raheem Sterling from Manchester City on Wednesday and are assessing their next move after losing out to Barcelona in the race to sign Raphinha from Leeds United.
However, ESPN reported last month that Tuchel is keen to encourage Chelsea to make a move for Robert Lewandowski if a deal is possible. Bayern Munich are reluctant to let the player leave, and his preference is thought to be a move to Barcelona.
Meanwhile, United continue to insist that Ronaldo is not available for transfer. New boss Erik ten Hag claims the Portugal forward is part of his plans for the upcoming season, and Ronaldo's options to leave appear to be narrowing after PSG distanced themselves from speculation they would make an offer.
Adriana Garcia contributed to this report.