Non-league Lincoln have been rewarded for their historic giant-killing exploits with a possible quarter-final trip to Arsenal.
Holders Manchester United face a trip to Premier League leaders Chelsea in the most high-profile last-eight tie.
Tottenham host League One Millwall in a London derby, while Middlesbrough welcome the winner of next week's fifth-round replay between Huddersfield and Manchester City.
Lincoln became the first non-league team to reach the competition's quarter-finals for more than a century when they stunned top-flight Burnley on Saturday.
They meet the winner of Monday's night's fixture between their National League rivals Sutton and 12-time cup winners Arsenal.
"It's a great draw," said Lincoln manager Danny Cowley.
"We wish Sutton all the best, I genuinely hope they can do it. If they can, then fantastic. If not, then we'll obviously play Arsenal.
"It's a win-win – we either play Sutton for the opportunity to be in the semi-final of the FA Cup or we have a fantastic tie away at Arsenal."
Ties are due to be played on 10-13 March 2017.
FA Cup sixth-round draw
Chelsea v Manchester United
Middlesbrough v Huddersfield or Manchester City
Tottenham Hotspur v Millwall
Sutton United or Arsenal v Lincoln City