Raheem Sterling will not discuss his future at Liverpool until the end of the season, according to Sky Sports sources.
Javier Hernandez has described his situation at Real Madrid as "frustrating" after playing 590 minutes for the club.
The 31-year-old has signed a three-year deal at the KC Stadium as the Tigers continue to prepare for their return to the Premier League.
McGregor told Hull's website: "I'm really excited to be here and I'm happy that everything is done now.
"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."