The 35-year-old is expected to serve as the club's third-choice stopper, behind David de Gea and Sergio Romero in the pecking order.
"Moving to Manchester United has been a dream of mine for as long as I can remember. This was an opportunity that I couldn't let pass and I'm thrilled to become a part of such a historic club," Grant told United's official website.
"I've enjoyed every moment of my career so far and I'm now ready to continue learning and improving alongside some of the best players in the game. But at the same time, I intend to use my experience to help the team whenever possible."