The player's desire is to return to the Italian club, who he refers to as his "family".
The Hammers beat Everton away and Man United at home with Rice playing a big role in both victories.
The west Londoners are prepared to offer Rice a deal worth £40,000-a-week as Maurizio Sarri targets a strengthening of his engine room.
Rashford's future has been the subject of much speculation at Manchester United and he has failed to cement a regular starting spot in Jose Mourinho's side.
Rashford has a contract at United until 2020 with the club having the option to extend that by another year, but any potential deal would hinge on the club being able to guarantee first team football.
Benjamin Netanyahu avoids early election as key ally stays
Will we stop making the murder of Jews in Israel worthwhile? Bennett responded to Netanyahu in his announcement on Sunday morning. It could also describe the latest chapter in a coalition crisis that threatens to bring down Benjamin Netanyahu's government.
The Portuguese coach made it apparent in no less terms over the summer that he wanted to sign a new centre-back, and now he also also wants more cover and competition in midfield.
This story could gather pace in the coming weeks as the winter transfer window draws ever closer, with a huge bidding war for Rice looking like a distinct possibility.
He has played three friendlies for the Republic of Ireland and has been approached about representing England, the country of his birth.
One club that may give him that option is Juventus, according to a report by Paul Joyce of the Times, who writes that the Bianconeri are "compiling a dossier" on the 21-year-old and are taking out "checks on the England striker's character and background".
They report that Real plan an audacious '£90m double raid' with Rashford and Eriksen in their sights.