The semifinal draw, for games that will take place on April 21 and 22, took place immediately after Chelsea beat Leicester City 2-1 in extra time in the last of the quarterfinals on Sunday evening.
United secured their place in the last four when they beat Brighton and Hove Albion at Old Trafford on Saturday thanks to goals from Romelu Lukaku and Nemanja Matic.
They will have the added challenge of facing Tottenham at Wembley, where Spurs have played all of their home games this season while waiting for the completion of their new stadium.
Before the draw, Tottenham defender Jan Vertonghen said the location would give his side an advantage, saying: “It definitely helps us because we have played well there.”
Tottenham have not lost at Wembley in the Premier League since August, when Chelsea repeated last season’s FA Cup semifinal success.
On April 22, the Blues will take on a Southampton side that beat the last remaining lower league team to get into the final four, with Saints having earned a 2-0 win over League One side Wigan Athletic on Sunday.
The two sides have met just once in the current campaign, when Chelsea secured a 1-0 victory through a Marcos Alonso goal.