Bobb has fallen down the pecking order at City after the January arrival of Antoine Semenyo from Bournemouth.
“I know there are rumours, I know there are talks,” manager Pep Guardiola said on Saturday after City’s win over Wolves.
Asked if Bobb wants to leave the club, Guardiola replied: “I think so.”
Bobb, who has also had interest from Bundesliga side Borussia Dortmund, has featured 15 times for City this season without scoring.
He last played on 17 December against Brentford in the Carabao Cup, when he hobbled off injured inside the opening 20 minutes.
Fulham have lost only one of their last seven Premier League games under Silva and sit seventh in the table following their 2-1 win over Brighton on Saturday.
Bobb has featured six times for Norway this year, helping them qualify for this summer’s World Cup, and would unite with compatriot Sander Berge at Craven Cottage.
He missed almost the whole of last season after fracturing a bone in his leg during training in August 2024.