The Red Devils have been quoted £70million to land the winger, 18, in what will prove a controversial move after Sancho’s time with neighbours Manchester City.
The England star has been a hit in Germany since going to Dortmund for £8m in August 2017.
Sancho spent nearly 2½ years in City’s academy after moving from Watford.
The club wanted to keep him but could not give him the assurances he wanted over first team playing time so moved to Germany.
He made 12 appearances scoring once last season with four assists.
This season he has been flying with eight goals and ten assists from 28 appearances.
Sancho, who is 19 next month, is enjoying life in Germany where he has become a hero with the Dortmund fans but a big move to Manchester United could prove difficult to resist.
However, City do have a 15 per cent sell-on clause in their original deal with Dortmund.
And if Dortmund do accept a bid from other club, the Etihad outfit would have the chance to match that offer if they wish.
Sancho, who played in Dortmund’s 3-0 defeat to Spurs at Wembley on Wednesday, certainly fits United’s traditional model of exciting young attacking players.
They could do with Sancho now after Solskjaer revealed yesterday that Jesse Lingard and Anthony Martial will be out for up to three weeks.
LIngard (hamstring) and Martial (groin) both came off halfway through the club’s 2-0 home defeat to PSG in the Champions League on Tuesday.
Their absence in the second half undoubtedly lead to United being overrun with neither Alexis Sanchez or Romelu Lukaku stepping up.
They will need to now starting with Chelsea at Stamford Bridge in the fifth round of the FA Cup on Monday night.
Teenagers Tahith Chong and Angel Gomes will be waiting in the wings if they don’t.
Interim boss Ole Solskjaer said: “There is a chance now for anyone who plays, whether Alexis, Romelu, Chongy or Angel.
"Anthony and Jesse have done really well for us, they're vital but then again with Rom and Alexis, they've got different skill sets and will be able to show what they can do. So we need to gel a team together for Monday.”
Solskjaer has been left in an odd position this week trying to pick the team and himself up from a first defeat under his charge after ten wins and a draw.
He said: “You don't panic because you've lost a game, you don't change everything, it doesn't change the mindset at all, we need to keep working at the way we play football.
“We want to go down to Chelsea, then we want to have Liverpool here and impose ourselves on those games.
“I hate losing games, the boys hate it, they've been down but there's no point dwelling on it. “There's aren't big games against good teams coming up and we need to dust ourselves down.
“You've got to bounce back at this club. You always have challenges and away against Chelsea is tough.
"Let's make sure during the next two days in training that we're ready on Monday.
"We know there was a spell against PSG, 7,8 10 mins we lost our way apart from that we weren't too bad.”
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