The semi-finals are due to be played on the week commencing January 6, 2014 with the second legs on the week beginning January 20.
United booked their spot in the last four with a 2-0 win at Stoke City with West Ham scoring twice late on to complete a 2-1 over managerless Tottenham at White Hart Lane.
An early second half goal from the returning Emmanuel Adebayor looked to have sent Spurs through to last four of the competition.
Manuel Pellegrini's City eased through on Tuesday night with a 3-1 win at Leicester with Sunderland upsetting Chelsea 2-1 at home after extra-time.
Both Manchester clubs will fancy their prospects of meeting in the final with United visiting the Stadium of Light for the first leg.
City will hope to build up a healthy lead to take with them to Upton Park after the first leg at the Etihad Stadium, a venue where they have been scoring goals for fun this season.