The 26-year-old's future has recently been the subject of much speculation, with a number of clubs believed to be interested in securing his signature this summer.
Hancko said earlier this month that he had held talks with new Liverpool boss Arne Slot but admitted that a switch to Anfield in the current window was "unlikely".
According to The Athletic, Man United are now expressing an interest, with the Red Devils identifying him as potential option to feature at left centre-back or left-back next season.
The report claims that Hancko would not be regarded as a starter if he moved to Old Trafford, but he would provide valuable cover in a position which has an issue for the Red Devils last term.
Man United 'exploring' deal for Hancko
The defender made the move to Feyenoord from Sparta Prague in 2022, and he has represented the Dutch giants on 93 occasions over the last two seasons, scoring 11 goals and registering eight assists.
Hancko enjoyed a strong 2023-24 campaign for his Dutch club, managing seven goals and three assists in 48 appearances, featuring as both a centre-back and a left-back.
He was then in action for Slovakia at Euro 2024, making four appearances at the competition, including the full 120 minutes of his side's 2-1 loss to England in the round of 16.
Hancko has another four years left to run on his contract, having only penned an extension earlier this year, but it has been suggested that he could be available for around €35m (£29.5m).
Shaw has returned to fitness for England at Euro 2024, but there are ongoing concerns surrounding Tyrell Malacia, who missed the entire 2023-24 campaign due to a knee injury.
Malacia is still battling to overcome a knee problem
Malacia made 39 appearances for Man United during the 2022-23 campaign after arriving from Feyenoord, but he missed last season due to a knee injury which required two separate operations.
The 24-year-old has not trained with the Red Devils this week, and it remains to be seen whether he is involved in his side's preparations for the new campaign.
Man United had huge problems at left-back last season due to injury problems, with Shaw also struggling with a number of fitness issues, and the Red Devils are said to be keen to bring in another player in that area of the field during the summer market.
Bayern Munich's Matthijs de Ligt, Everton's Jarrad Branthwaite and Lille's Leny Yoro are all currently being linked with moves to Old Trafford to improve the centre-back options at the club, so it is unlikely that Hancko would receive much football in that area of the field.
However, the Slovakian could be a valuable option at left-back, and his impressive goalscoring record is likely to make him even more attractive to the Red Devils.