Metro, as quoted by the Daily Mirror, is reporting that the Red Devils are interested in signing the 22-year-old in a bid to improve their options at the back.
Manchester United manager Jose Mourinho is looking to sign Gaya to replace out of favour Italy international Matteo Darmian.
According to the same story, the former Torino defender is set to move to Serie A champions Juventus at the end of the season following a difficult campaign at Old Trafford.
Gaya has scored four times in 106 games in all competitions during the past four seasons after he made his debut in the Europa League in 2013.
The same article goes on to suggest that Arsenal and Chelsea FC are also interested in the 22-year-old full-back ahead of the summer transfer window.
Mourinho has relied heavily on England veteran Ashley Young throughout the current campaign given his reluctance to start former Southampton defender Luke Shaw.
Manchester United lost 2-1 to Spanish side Sevilla in the second leg of their Champions League last-16 clash at Old Trafford on Tuesday night.
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