No fee was disclosed although media reports said United had agreed a deal of about 25 million pounds ($36.44 million) with the La Liga side.
"Eric is a young central defender with great natural talent. He has progressed well to date and has the potential to become one of the best around," former Chelsea manager Mourinho, who replaced Dutchman Louis van Gaal last month, said.
"We look forward to working with him to help nurture that raw talent."
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Bailly has 15 caps for Ivory Coast and was part of their 2015 Africa Cup of Nations-winning squad.
He played 47 times for Villarreal after joining in 2015 from Espanyol, impressing in a back four that conceded only 35 goals in 38 La Liga matches last season and reached the Europa League semi-finals, losing to Liverpool.
"I want to progress to be the best that I can be and I believe working with Jose Mourinho will help me develop in the right way and at the right club," said Bailly, who will likely partner England's Chris Smalling in central defence next season.
The capture of Bailly marks the first of an expected flurry of signings by Mourinho as he seeks to restore the fortunes of the 20-times English champions who won the FA Cup last season but finished fifth in the Premier League.
United, who will play in the Europa league after missing out on Champions League place, are also favourites to sign Sweden striker Zlatan Ibrahimovic while Juventus midfielder Paul Pogba has also been linked with a move to Old Trafford.
(Reporting by Martyn Herman; Editing by Ken Ferris)
