The live TV draw saw the Reds picked last out of the hat and given an away tie at the Academy Stadium.
Vicky Jepson’s side secured their place in the last eight with a 2-0 win over Millwall Lionesses at Prenton Park on Sunday.
Second-half goals from Kirsty Linnett and Rinsola Babajide ensured a fourth straight victory for the Reds.
Quarter-final ties are due to be played on Sunday March 17.
Full draw
Reading Women v Manchester United Women
Aston Villa Ladies v West Ham United Women
Durham Women v Chelsea Women
Manchester City Women v Liverpool FC Women