The Germany goalkeeper is available from Barca this summer, with management wishing to go with Wojciech Szczesny and new arrival Joan Garcia as their keeper options for the new season.
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Ter Stegen is reluctant to leave, but he could be offered the chance to move to England and City.
AS says City see the veteran keeper as a replacement for Ederson, who could now accept a move to the Saudi Pro League.
Ter Stegen, 33, has a deal to 2028 with Barca, but could be granted a free transfer with the Catalans eager to clear his wages off their books.