Hart has been cast aside by new manager Pep Guardiola and has not featured in the Premier League this season for City.
The England No 1 looked to have played his final game for the club last night in a teary 1-0 win over Romanian side Steaua Bucharest.
And now City director Begiristain has confirmed the goalkeeper’s imminent departure, revealing they are in the process of finding Hart a new club.
"We will try to find a good solution for everybody,” Begiristain said.
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Manchester City have revealed Joe Hart will be leaving the club before deadline day
“We now have one week to make him happy and find a club where he has a real chance.
"We will try and help him because he deserves it."
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Hart has made 266 Premier League appearances for City since signing from Shrewsbury in 2006 and has twice been named in the PFA Team of the Year.
City have already signed his replacement, bringing in Claudio Bravo from Barcelona on a four-year deal.