The former Manchester United manager has been out of football since his sacking in December 2021.
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Solskjaer said at his presentation: "It's great to be here.
"It's very nice to see so many people who care about this fantastic football club.
"I've had a very nice first meeting with the members of the board.
"I'm really looking forward to this - it's fantastic to be in this beautiful city."
Besiktas axed Giovanni van Bronckhorst last month. The Black Eagles currently sit sixth in the Super Lig table.