On the back of talks being held between Blues owner Todd Boehly and super-agent Jorge Mendes, United insisted that the 37-year-old was not for sale.
While that stance remains the same, reports emerged over the weekend that the Portuguese superstar had requested a transfer away from Old Trafford.
The veteran attacker is said to want to play Champions League football during the latter stages of his career, with at least a year in the Europa League having little appeal to him.
On Monday afternoon, it was suggested that Chelsea were contemplating whether to put forward a formal proposal for Ronaldo over the coming weeks.
Nevertheless, according to The Mirror, United would rather not entertain the prospect of the legendary figure moving to one of their Premier League rivals.
Despite 16 points separating the two clubs in 2021-22, new United head coach Erik ten Hag is currently in the process of making significant changes to his first-team squad.
Although a certain level of ambition is being shown by the former Ajax boss and club officials, Ronaldo is allegedly unhappy with the current state of play.
Ronaldo had been due to return to first-team training on Monday, only for United to be accepting of his absence due to family reasons.
However, Ronaldo has reportedly been told that he will be expected to feature on United's upcoming pre-season tour of Thailand and Australia.
The Portugal international is coming off the back of a strong first year back at United, contributing 24 goals from his 38 appearances in all competitions.
Nevertheless, it does not appear that he has a lengthy queue for his signature, The Mirror adding that Bayern Munich have no interest in a deal.
A different report has indicated that Paris Saint-Germain place themselves in the same category, although Napoli are said to be monitoring the situation.
Ronaldo has one year remaining on his contract, while United possess the option of extending his terms through to the end of 2023-24.