The England international has just less than a week to secure a move away from the Citizens after being made surplus to requirements by new manager Pep Guardiola
Hart has become City's third-choice goalkeeper since the arrival of Pep Guardiola, with new signing Claudio Bravo and Willy Caballero both ahead of the England international in the pecking order.
The 29-year-old was given an emotional farewell from City fans as he seemingly played his last game for the club in a 1-0 win over Steaua Bucharest in the Champions League play-offs on Wednesday.
While Everton have ruled out a move for Hart, Begiristain says City are aiming to find the right solution for the future of the long-serving keeper.
"I think we're going to try to find a good solution for everybody," Begiristain said.
"Now we have one week and we have to make him [Hart] happy and if he can find a solution and find a club where he has a real chance to play regularly.
"So we're going to try to help him, because he deserves the best answer from us."