The 30-year-old goalkeeper has completed his move to the Istanbul-based club after rejecting a new deal to stay to stay in Scotland with Ally McCoist's side
The 30-year-old was one of nine players who rejected their contracts being transferred to the Rangers Newco this summer, bringing an end to an 11-year affiliation with the club.
McGregor, though, is excited about the prospect of testing himself in another country.
"I became free agent after Rangers were relegated. Besiktas showed interest in me and I came to Turkey," McGregor told BJK TV.
"I realise I came to a big club with traditions. This club's supporters give enormous support to their club. This will be a new challenge and goal for me, I feel I'm ready for it."
"I give everything for my team on the pitch. I cannot stand to lose. I will do my best for Besiktas.
"I took an opportunity to watch our supporters from videos, I know how enthusiastic the Turkish fans are."
The Scotland international enjoyed a successful time with his former club, winning three league titles and three Scottish Cups, also racking up 20 caps for the national side.