The Blues defeated another championship side Luton Town 3-2 on Wednesday while Middlesbrough saw off Premier League opposition in the shape of Tottenham Hotspurs 1-0 after extra time.
The winner of the final fifth round tie between Nottingham Forest and Huddersfield on Monday will face Liverpool.
Crystal Palace will take on Everton while Manchester City travel to the Saint Mary’s Stadium to take on Southampton.
The quarterfinal ties will take place on the weekend commending March 19-20.
This season’s FA Cup will be played on May 14 at Wembley.
Leicester City beat Chelsea 1-0 to lift last season’s FA Cup for the first time.
FA Cup quarter-final draw:
Crystal Palace v Everton
Middlesbrough v Chelsea
Nottingham Forest or Huddersfield v Liverpool
Southampton v Manchester City
