Goalkeeper Given is warming the bench at Eastlands after losing his place to Joe Hart and the 34-year-old has been linked with a move away.
Roma were interested during the summer and Trapattoni, Given's international manager, believes he should consider the Italian top flight to reignite his career.
Frustration: Given has not played a Premier League game for City this season
He said: 'Me and Marco (Tardelli, Ireland assistant boss) have spoken with some Italian teams. Shay is kept in high regards by many clubs but goalkeeper is a difficult position.
'One or two Italian teams need a goalkeeper. There is interest in him. There are clubs in Italy with inferior goalkeepers. If he has the opportunity, maybe he will ask to leave City.'
Given was linked with a move back to Newcastle, the club he left to join City, after Steve Harper was ruled out with a shoulder injury at the weekend.
But Magpies manager Chris Hughton poured cold water on talk of a Tyneside return.
He said: 'I can understand that one and it is a very emotional one as well, but no, I can categorically say that won't be happening.'
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