The Italy international winger joined Torino from Fiorentina in the summer of 2012 and has been impressive in the Serie A this season.
The Daily Star reports that United have been keeping tabs on Cerci for a number of months and are considering making a move for the winger in January.
But United now face competition from Arsenal as manager Arsene Wenger is ready to make a January move for the winger.
Serie A champions Juventus are also reportedly interested in the winger, valued at £16million by Torino.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.