The England international has been on the Red Devils transfer radar for several years and he's now back to full fitness at the Hill Dickinson Stadium.
Moyes has brought him straight back into Everton's starting XI in 2026/27 and the 24-year-old's form has secured a recall to Thomas Tuchel's latest England squad.
Everton's stance over what comes next appears to be clear with the club unhappy with United's previous 'insufficient' transfer offer in 2024.
The Toffees are under no pressure to sell, after Branthwaite signed a contract extension until 2030 less than a year ago, and Moyes believes they can keep him
"It was a brilliant bit of recruitment by Everton - also putting him out on loan at the right moments.
"Unfortunately, he's missed a couple of bits of seasons through injury. There's no doubt he's a great footballer.
"He was someone who, when I came in, I was determined the club should keep hold of."