Stones, 31, joined from Everton in 2016 for £47.5 million and has won 17 trophies during a decade at the Etihad. However, persistent injuries have severely limited his involvement this season.
He has made 15 competitive appearances in total and has not featured in the Premier League since early December.
The Athletic journalist Sam Lee has also reported that Nathan Aké could leave when his contract expires in 2027, though nothing is certain on that front. The Dutch defender has been a regular under Pep Guardiola in the second half of the season after a difficult start, making 14 league appearances, four from the start.
City have already reinforced their defensive options significantly. Marc Guehi arrived in January and Brazilian prospect Vitor Reis is due to return from his loan at Girona this summer, making the continuation of Stones and potentially Aké increasingly difficult to justify.
Everton are among the clubs monitoring Stones ahead of a potential free transfer, which would mark a return to his boyhood club a decade after he left Goodison Park.
Manchester City are second in the Premier League, six points clear of Arsenal with a game in hand. The two sides meet on Sunday in a fixture that could decide the title.