The glamour fixture sees Manchester United manager Jose Mourinho returning to Stamford Bridge with the holders and 12-time FA Cup winners to face his old club Chelsea, the Premier League leaders.
Lincoln, who play in the fifth tier of the English game, could feature in a unique last-eight tie as a reward for their sensational 1-0 win at Premier League Burnley on Saturday.
Should the two National League clubs meet it would guarantee the outlandish feat of a minor league club making it to the semi-finals at Wembley Stadium for the first time.
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First, Sutton have the monumental task of winning their own dream tie at home to Arsenal in the fifth round on Monday.
Lincoln manager Danny Cowley, who watched the draw at the club's Sincil Bank ground surrounded by cheering fans, called it a "win-win" draw for the team who top the National League.
"It's great. We wish Sutton all the best tomorrow - I genuinely hope they can do it. If they can, then fantastic," he said.
"For us, it's a win-win, we either play Sutton for a place in the semi-final or we have a fantastic tie away at Arsenal.
"It's completely surreal. I feel like we're going to wake up at any moment."
Sutton are 17th in the National League, 104 places below Arsenal in English football's league pyramid.
Lincoln face the prospect of a visit to either Sutton's ramshackle 5,000-capacity Gander Green Lane ground or Arsenal's 60,000-seater citadel, the Emirates Stadium.
When Chelsea's most successful manager Mourinho, who won the Cup with the Londoners 10 years ago, went back there in October he suffered the most humiliating defeat of his United reign as Antonio Conte's men won 4-0 but that was their last league loss.
After United won a hard-fought contest 2-1 at Blackburn Rovers on Sunday, Mourinho could look on a resurgence that has seen his side carve out the longest unbeaten league sequence -- 16 matches -- in any of Europe's top five leagues this season.
Tottenham Hotspur, who eased their way into the last eight with a Harry Kane hat-trick seeing off Fulham 3-0 at Craven Cottage, next host third-tier Millwall, who knocked out Premier League champions Leicester City on Saturday.
Spurs fans looking for a happy omen quickly latched on to the fact that when they won the Cup in 1966-67, they beat Millwall in the third round on the way to picking up the trophy.
Middlesbrough could host an all-Premier League tie against Manchester City if Pep Guardiola's side, as expected, beat Huddersfield Town at the second time of asking following Saturday's goalless draw at Kirklees Stadium.
Two years ago, City were knocked out 2-0 by then second-tier Middlesbrough at the Etihad Stadium in the fourth round.
Sixth-round Draw (Premier League unless stated)
Chelsea v Manchester United
Middlesbrough v Huddersfield Town (II)/Manchester City
Tottenham Hotspur v Millwall (III)
Sutton United (NL)/Arsenal v Lincoln City (NL)
Ties will be played from March 10-13
(Writing by Ian Chadband; Editing by Ken Ferris)
