There was no immediate confirmation from City, who finished third in the Premier League this season under Spanish manager Pep Guardiola.
The fee would be close to the world record for a goalkeeper, with Juventus paying a reported 40.2 million euros when they signed Gianluigi Buffon from Serie A rivals Parma in 2001.
Benfica said also that half of the fee would be paid to unidentified 'third parties', an element that could slow completion of a deal.
Third party ownership, where part of a player's economic rights are owned by private investors, is not allowed under Premier League rules and that will need to be resolved before the player can be registered.
