According to MailOnline's Robert Summerscales, United had been considering making a January bid for the 22-year-old. However, it is said he may not be able to pass a medical after suffering a recurrence of a leg injury.
Goretzka's Schalke contract expires next summer, meaning he can negotiate a pre-contract agreement with foreign clubs from Jan. 1.
Schalke announced via Twitter on Wednesday that there is no date set for his return to action (h/t Summerscales): "Leon Goretzka has suffered a recurrence of the injury in his lower leg. In order to put these continuous and worsening pains to bed, Leon will undergo examinations over the coming days. Until then, it is unclear as to when Leon will be fit to train and play once again."
United are not the only major European club to have been linked with Goretzka. Sky Sports News reported on Saturday that Barcelona, Juventus, Tottenham Hotspur and Arsenal are also interested in the Germany international.
However, Marca's Chris Winterburn believes the 22-year-old is set for a move to Bayern Munich when his contract expires:
It would be little surprise if Goretzka were to snub interest from elsewhere in order to stay in Germany and join the Bundesliga champions.
But it would also come as no shock if the likes of United, Barcelona, Juve and Arsenal pushed hard to beat Bayern to his signature despite his injury problems.
Goretzka is a central midfielder of supreme potential and talent, and he is improving every season.
Long seen as an accomplished deep-lying midfielder who can protect the defence, he has added an attacking prowess to his game in the past 18 months that was not previously apparent.
In the 2017-18 campaign, he has netted four goals in 11 appearances after returning five in 30 last term, along with three assists.
Goretzka looks to have the talent to become one of the world's best box-to-box midfielders, so it is little surprise a number of clubs are after him, especially as he could be available for free.