The first legs will be played on Feb. 20, with the return games on Feb. 27.
Manchester United will play against Belgian champion Brugge in the round of 32 of the Europa League.
Monday's draw also paired Arsenal, last year's runner-up in the second-tier competition, with Greek club Olympiakos.
Brugge dropped into the Europa League after finishing third in its Champions League group, which included Paris Saint-Germain and Real Madrid. It drew 2-2 at Madrid.
Ajax, which also transferred across from the group stage of the Champions League, was drawn to play Getafe, with the first leg in Spain.
Three seasons ago, Man United beat Ajax in the Europa League final to secure a place in the Champions League.
Other teams switching across as third-place finishers in their Champions League groups include Benfica and Shakhtar Donetsk, who were drawn to play each other.
Inter Milan was paired with Ludogorets Razgrad, and Salzburg will face Eintracht Frankfurt.
Arsenal will play the first leg against Olympiakos in Greece. Record five-time champion Sevilla was paired with Romanian club Cluj, and Roma will play Gent.
The first legs will be played on Feb. 20, with the return games on Feb. 27.
The May 27 final will be in Gdansk, Poland.
