The 31-year-old has signed a three-year deal with Steve Bruce's side, while the move is subject to international clearance.
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"I’m really excited to be here and I’m happy that everything is done now," McGregor told the club's official website.
"I can’t wait to get started here. This is a great chance for me to play in the Premier League and I’m looking forward to the season ahead.”
The Scot made 26 league appearances for Besiktas last season, concedeing 39 goals and keeping five clean sheets.
McGregor's arrival at the KC Stadium makes him the club's fifth permanent signing of the summer following the arrivals of George Boyd, Maynor Figueroa, Curtis Davies and Ahmed Elmohamady.