The Athletic reports Winks has been discussed by senior figures at Old Trafford.
The midfielder was handed a start in the FA Cup against Middlesbrough on Sunday but was again withdrawn before the hour mark after suffering an ankle injury in the first half.
His lack of playing time has reportedly alerted both United and City, who could make moves before the end of the month.
At City, Fernandinho turns 35 in May while skipper David Silva has already confirmed he will be leaving the club this summer.
City are admirers of Winks, whose ability to dictate play from a deep midfield position has alerted club scouts.