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.
Although the Red Devils were drawn as the home side in the April 21 tie, the game will be played at Wembley, the stadium that Tottenham, who got the better of fellow Premier League side Swansea City in their quarter-final, have called home this season.
United and Spurs have beaten each other already in the league this term, with Jose Mourinho's side winning 1-0 at Old Trafford before falling to a 2-0 defeat at Wembley in January.
Last year's runners-up Chelsea needed extra-time to edge past Leicester City at the King Power Stadium on Sunday. Jamie Vardy struck for the Foxes to cancel out Alvaro Morata's first-half goal, but substitute Pedro emerged to net the decisive goal for Antonio Conte's team.
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.