The midfielder was linked to the Premier League champions at the start of the season and rumours of a January transfer have intensified in recent days following comments by Roma coach Zdenek Zeman.
Ahead of De Rossi's return from a three-match ban for violent conduct, Zeman indicated De Rossi could be surplus to requirements.
"De Rossi is not a problem for me but he has under-performed and could have done better this season," Zeman told Rai1 television channel.
"He has not integrated himself into the squad."
Reports claim De Rossi has struggled to adapt to Zeman's preferred 4-3-3 formation and the pair are reported to have clashed after a recent defeat to Juventus.
TalkSport also claimed De Rossi, who has come up through all the ranks at his hometown club, flew to London recently for talks with representatives with City.
Roma currently sit sixth in the league, nine points behind leaders and champions Juventus.