"He has progressed well to date and has the potential to become one of the best around.
"We look forward to working with him to help nurture that raw talent and fulfil his potential.
"Eric is at the right club to continue his development."
Bailly becomes only United's second defensive signing from their past three transfer windows.
Right-back Matteo Darmian joined the club from Torino last summer, with Daley Blind the previous defender to arrive at Old Trafford back in September 2014.
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