Personal terms have been thrashed out on a three-year contract which will begin once Bruce’s top-flight new boys gain international clearance.
It means McGregor, 31, is set to make his debut against Chelsea on the opening day of the season.
He said: “I’ve had a few chats with the manager here and I really had belief and confidence in what he said about the club.
“The chance to play in the Premier League is massive.
“I got a call a while ago about the interest from Hull and things progressed from there.
“Holidays stalled things a bit, as they do at this time of year, but everything is done now and I’m really happy to be here.
“I kept an eye on what was going on here while I was in Turkey. I even watched the last few minutes of Hull’s game with Cardiff on the final day of the season — it was exciting.
“Now we have the excitement of the Premier League to look forward to. We play Chelsea in our first game, so it’s one we are all looking forward to.”