Rojo has not joined United on their US tour, despite Argentina having exited the World Cup exactly a month ago, giving him more than the requisite three-week post-tournament break.
Although Rojo signed a new deal in March until 2021 with the option of an extra year, that was done chiefly to protect his market value.
As well as Everton, Wolves are understood to have expressed interest in Rojo, but Goodison Park is a more realistic destination for the 27-year-old.
Everton have also opened contract talks with England hero Jordan Pickford in a bid to head off interest from Chelsea.
The keeper was the biggest star of England’s unlikely charge to the World Cup semi-finals where his form attracted the interest of the wealthy London club.
Everton have remained relaxed about the interest in the keeper they snapped up from Sunderland for what now looks like a bargain £30million last sumnmer.
But the reality is that Pickford’s summer has led to the Goodison club scrambling to offer him an improved, longer deal.
New Toffees boss Marco Silva, meanwhile, is planning on having Pickford in his starting line-up for the opening clash of the new season at Wolves a week on Saturday.
As with all Everton’s World Cup contingent, Pickford has been given a strict three-week holiday period following the end of the tournament, meaning he is not yet back on Merseyside.
He will be absent when Silva winds up his pre-season against Valencia this weekend, with back-up Maarten Stekelenburg continuing to deputise.
Stekelenburg signed a new two-year deal with Everton yesterday, keeping him at the club until 2020, as Silva looks to use his experience to help in the development of Pickford.
But Pickford has been on a regimen from Everton’s fitness experts to make sure he is in game shape in time for the big kick-off on August 11.
Meanwhile, Silva is closing in on his second big summer signing following the £44 million capture of Watford’s Richarlison.
Barcelona left-back Lucas Digne was back in the Catalan city last night tying up lose ends over a £19m switch to Goodison.
Everton veteran Leighton Baines is entering the final year of his contract at Goodison and Digne, 25, is seen as his heir.