It's claimed the Dutchman is fancied by compatriot Ten Hag, who wants to bring him in as a coach.
The news could be potentially fatal for recently relegated Burnley, with Van Nistelrooy heavily tipped to replace Vincent Kompany at Turf Moor.
But it may also prove detrimental for current United coach Benni McCarthy, whose contract with the three-time European champions is set to expire this summer.
McCarthy was hired when Ten Hag arrived in England two summers ago, but the South African hasn't been offered an extension, with his future at the club in doubt.
Ruud van Nistelrooy has been 'approached' by Man United. (Image: Getty)
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Brought in as an attacking coach, McCarthy is believed to have inspired Marcus Rashford's 2022/23 campaign, which saw the local hero register 40 goals in 56 appearances across all competitions.
Then there's Rasmus Hojlund, who might not have reached the heights expected of the £72million signing, but it's claimed
McCarthy has had a positive influence on the Dane.
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It comes amid a busy period for Ten Hag, who is said to have angered INEOS officials following his recent appearance on Dutch television.
The ex-Ajax chief let slip that members of the United hierarchies had met him while on holiday in Ibiza to discuss his future, including their talks with alternative managerial candidates.
It's said the admission from the United manager did not go well behind scenes, as Sir Jim Ratciffe and co. prep for their first summer in charge.
United will begin their Premier League campaign in less than two months time, taking on Fulham in the opening game on Friday August 16.