Villa had agreed an £8m fee with United, but walked away on Monday after failing to agree personal terms.
Everton also tried to sign Cleverley on transfer deadline day, but failed to strike a deal, which then saw Villa resurrect the move. They are believed to have a deal in place to make the loan permanent.
"It's a team that plays the style of football I like," Cleverley added. "They play with high energy, they pass and move and there are a lot of good young players. The manager told me the way that he'd like me to play and it all sat perfectly for me. I'm very happy to be coming down to play for Villa."
Villa boss Paul Lambert added: "Tom brings big game experience to the club, he's played international football for England and he still has great potential to kick on in his career.
"There's a lot of good football ahead of him if he performs the way we think he can perform and we're delighted to have been able to bring him in to join the group."
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