After much debate over who would be the Blues' number one, Mancini plumped for England international Joe Hart on the opening day of the season against Tottenham Hotspur.
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Hart produced an outstanding display in the goalless draw with Given left to watch on from the bench.
The Irishman had indicated that he would not settle for a substitute role at Eastlands this season, alerting a number of clubs to his availability, including Arsenal boss Arsene Wenger.
The Emirates Stadium supremo is believed to be on the lookout for a long-term first choice stopper with question marks continuing to remain over both Manuel Almunia and Lukasz Fabianksi, and has strongly been linked with a move for Fulham's Mark Schwarzer.
And Mancini has revealed that, although he hopes Given will stay, he will not prevent the 34-year-old to depart.
“I hope he can stay. But I respect his decision if he wants to leave,” he said.
Any potential move could depend on Schwarzer’s future, with Cottagers' boss Mark Hughes believed to be mulling a move for Given should the Australian join the Gunners.
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Schwarzer sat out Fulham’s opening day draw against Bolton amid speculation over a £2.5 million move to the Gunners.
Given would provide an ideal replacement for Schwarzer at Craven Cottage, although Arsene Wenger has refused to rule out a direct move for Given.
In this goalkeeper transfer merry-go-round Mancini may refuse to sell to Arsenal in the same way that he reportedly refused to let Craig Bellamy join Spurs.
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The Italian considers Wenger’s men as rivals to City’s ambitions this season, meaning a deal with Fulham would be more likely.
However, Mancini has not given up hope of persuading Given to stay.
By Dave Peddie
