Holders Manchester United play Chelsea, and Millwall, who beat Premier League champions Leicester, play Tottenham.
HLincoln City will play Sutton or Arsenal in the FA Cup quarter-finals as reward for their stunning fifth-round victory over Burnley.
The fifth-tier club became the first non-league team in 103 years to reach the last eight with the biggest shock of the competition so far on Saturday.
Middlesbrough face Manchester City or Huddersfield, who drew 0-0 on Saturday.
“It’s a win-win,” Lincoln manager Danny Cowley said on BT Sport. “We wish Sutton all the best tomorrow, I genuinely hope they can do it.”
Sutton, who are the lowest-ranked team still left in the cup, play at home to the Gunners on Monday.
The replay between Manchester City and Huddersfield is provisionally set for Tuesday, 28 February at Etihad Stadium.
Full FA Cup quarter-final draw:
Chelsea v Manchester United
Middlesbrough v Huddersfield/Manchester City
Tottenham v Millwall
Sutton/Arsenal v Lincoln