The ties will be played at Wembley on the 18th and 19th of July.
Manchester United sealed their place in the last four on Saturday when they won 2-1 away at Norwich after extra time, as Harry Maguire struck with two minutes left to play against the hosts, who went down to 10 men late in the second half.
Chelsea went through with a narrow scoreline, too, as Ross Barkley was on target in the 63rd minute to secure a 1-0 win at the King Power Stadium on Sunday.
United and Chelsea have faced each other three times already this season, with Ole Gunnar Solskjaer's team coming out on top each time. The Red Devils opened their Premier League campaign with a 4-0 win over Frank Lampard's team before beating them 2-1 in the Carabao Cup in October. They then got the better of the Blues at Stamford Bridge with a 2-0 Premier League victory in February.
However, Chelsea's last FA Cup success came at United's expense, as the London outfit won the trophy for the eighth time in their history when they ran out 1-0 winners in 2018.
Arsenal progressed to the last four thanks to a late winner against Sheffield United on Sunday, with Dani Ceballos popping up with the decisive goal after Nicolas Pepe's first-half penalty was cancelled out by a David McGoldrick goal.
Manchester City secured their place in the the semi-finals with a 2-0 victory against Newcastle at St James' Park, with Kevin De Bruyne and Raheem Sterling on target.
Pep Guardiola's team are looking to defend their crown, having beaten Watford in the decider last year, but Arsenal are the FA Cup's most successful team, having won it 13 times in their history.
Manchester United are the second most frequent champions with 12 wins to their name, with their most recent coming in 2016. City, meanwhile, have won it six times each.
The FA Cup final is scheduled to be played at Wembley on August 1.