The two clubs had been negotiating over a deal for the past week and a final £27M fee was agreed for the 22-year-old who has had time at City heavily affected by several injuries.
ESPN report that City have also inserted a 20% sell-on clause and agreed matching rights on any future bids for the youngster.
Speaking on the move, Bobb said he is delighted to move down South to the London side who he cannot wait to get started with.
“It feels great to be here. I had a great day meeting everyone and I’m very excited.
“I’ve always known Fulham to be a good club, with good players and a great stadium. I spoke to Sander (Berge), my good friend, and I spoke to the Manager, and he explained what the system’s like, and how the lads are, and how the Club is.
“He seemed lovely, so it was an easy decision basically.”
Bobb has made 15 appearances for City this season but after the signing of Bournemouth’s Antoine Semenyo, he was unlikely to feature for the side again under manager Pep Guardiola.
Since his return from injury, the winger has made eight appearances for Norway, helping the side qualify for their first World Cup in 27 years.