Rangnick's deal is set to see him move into a consultancy role at Old Trafford after the end of the current campaign.
The German was appointed on Monday on a contract until the end of the season after the sacking of Ole Gunnar Solskjaer and will work in a consultancy role for a further two years.
United's hierarchy have resisted calls to send Solskjaer packing after last weekend's unthinkable 5-0 home thrashing by arch-rivals Liverpool in the Premier League.
The two teams will meet on Sunday at Manchester United’s cathedral, Old Trafford.
The injury cut short his game time, with the 23-year-old playing only a limited role with England during their run to the Euro 2020 final in July.
Kortney Hause's 88th-minute header put Villa on course for their first league win over United since 2009.
Rashford has returned to the club's Carrington training ground to continue his recovery from a recent shoulder operation, manager Ole Gunnar Solskjaer said.
Solskjaer says his talks with Paul Pogba have been "positive" amid reports linking the France midfielder with a move to Paris Saint Germain.
United finished third in the league in Solskjaer's first full season in charge in 2019-20, and also reached the semi-finals of the League Cup, FA Cup and Europa League.
The 33-year-old Spain international joined from Chelsea in 2014 and has made 273 appearances for United, scoring 51 goals.
Manchester United's gut-wrenching defeat on penalties by Villarreal in the Europa League final re-emphasised the magnitude of the rebuilding task for Solskjaer.
Solskjaer said he hopes the club's progress this season will convince Paul Pogba and Edinson Cavani to extend their stay at Old Trafford.
Bruno Fernandes and Edinson Cavani each scored twice as Manchester United took a huge stride towards the Europa League final.
Solskjaer tried to placate angry fans of Europa League semi-final opponents Roma, who have accused the Manchester United boss of disrespecting their club.
Solskjaer says giving up on the Premier League title is "not in our DNA" even though their pursuit of runaway leaders Manchester City appears hopeless.
After Mourinho's appointment as Manchester United manager in 2016, he made Pogba the world's most expensive player
Solskjaer said Rashford, 23, was available for the trip to Spain but might not be risked for the entire match.
United have 36 points from 17 games, three more than second-placed Liverpool before the blockbuster clash at Anfield.
Solskjaer’s side could find themselves in second place and within two points of leaders Liverpool with a win at Leicester City on Saturday.
Manchester United’s dismal home record in the Premier League has nothing to do with a mental block or the absence of fans, manager Ole Gunnar Solskjaer said.