The 28-year-old, who has entered the final year of his contract, is being sought by the Premier League outfits from London and the North West, according to talkSPORT.
Despite the interest, the Italy international aims to respect the Serie A club and eventually put pen to paper on a new deal.
"Everything is at an impasse," De Rossi has been quoted as saying. "We have discussed the issue in recent weeks and I am sure we will find a good solution for everyone.
"I will not put pressure on Roma. I never said I would sign for another club and that's true - that would not be just, considering I get on with those in charge."
City are reportedly considering to offer in the region of £8m for the playmaker.