The veteran was on loan with City last term, and did not play a single game this season.
But The Athletic indicates he is happy to come back for another season, and City would like to do a deal as well.
The 34-year-old would be cover for Ederson and Claudio Bravo, who are first and second choice respectively.
The source claims Carson would stay for another year and eventually move into coaching, as he is entering the final year or two of his career.
The veteran has a lot of Premier League experience and it is what City want in their backup keeper.