Boro must await the winner of Manchester City and Huddersfield, who shared a goalless draw on Saturday and face a replay.
But the Smogs have shown an interest in the competition, unlike Newcastle and Sunderland.
Which asks the question - if you have no intention of TRYING to win, why enter the competition in the first place?
Lincoln have become the first non-league team in 103 years to reach the last eight, and Sunday evening's draw saw them handed a mouthwatering trip to the Emirates Stadium or an all-National League meeting with Sutton.
Arsenal play Sutton at Gander Green Lane on Monday evening.
Jose Mourinho will return to Stamford Bridge as Chelsea host his Manchester United side, Middlesbrough will be at home to Huddersfield or Manchester City, and League One outfit Millwall go to Tottenham.
Lincoln shocked Burnley with a 1-0 victory at Turf Moor and the scorer of the winner, Sean Raggett, will be particularly pleased with the draw given he is an Arsenal supporter.
"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."
FA Cup holders Manchester United, meanwhile, are tasked with halting Premier League leaders Chelsea's charge towards winning the double. United were beaten 4-0 by Mourinho's former side when the teams last met in the league in October.
Tottenham's London derby against Millwall, who knocked out Premier League champions Leicester in the fifth round, will be the last FA Cup match at White Hart Lane before Spurs move out of their current ground at the end of the season.
They will be wary of Millwall, who have form for making the latter stages of the tournament, after losing to Manchester United in the 2004 final, before making a surprise run to the semi-finals in 2013.
FA Cup quarter-final draw
Chelsea v Manchester United
Middlesbrough v Huddersfield or Manchester City
Tottenham v Millwall
Sutton or Arsenal v Lincoln
Matches to be played on weekend of March 10-13