But the Italians face rivalry from at least one English club.
Whether City will be prepared to sanction a temporary or permanent move for the 34-year-old Irishman is unclear.
Given, who is on £100,000 a week, has expressed his dissatisfaction with being No 2 to Joe Hart at Eastlands and manager Roberto Mancini has said he would be willing to let him leave.
Frustration: Shay Given is unhappy asstandby to Joe Hart at Eastlands
But the City hierarchy have not taken the same line and are unlikely to sell to another Barclays Premier League club.
In August, Roma, coached by former Chelsea boss Claudio Ranieri,spoke about a £2million deal and also a possible swap arrangement.
Roma, who finished second to Jose Mourinho's Inter Milan in Serie Alast season, are currently fifth in the table, four points off aChampions League place.
They have reached the last 16 of this season' s Champions League and face Shakhtar Donetsk.
Given's Republic of Ireland manager Giovanni Trapattoni is a hugefan, and he is known to have mentioned the goalkeeper to Italian clubs.
Middle man: Giovanni Trapattoni has spoken to Italian clubs
The player has not hidden his frustration at his situation since the summer and would move abroad if that was his only option.
However, English clubs are also monitoring City's attitude towardsthe hugely experienced keeper who returns to Newcastle United for thefirst time on Boxing Day following his departure in January 2009.
City set sights on raid for Torres as Wolfsburg demand £38m for DzekoMancini tells bad boy: Lose the frown and everything will be super, MarioMANCHESTER CITY FC