As part of sanctions placed on the Blues by the UK government because of owner Roman Abramovich's links to Russia president Vladimir Putin, Chelsea are not allowed to sell tickets for the game.
Chelsea sold close to 600 tickets of their 4,620 allocation before the sanctions were imposed while Middlesbrough have exhausted theirs.
Chelsea feel their fans who can attend the game will be inadequate and want Middlesbrough's to be blocked as well 'to protect sporting integrity'.
A statement on the club's website reads: "Despite engaging in extensive discussions with the Office of Financial Sanctions Implementation (OFSI), the deadline to purchase away tickets has passed without appropriate amendments being made to the government licence which would allow a full allocation of Chelsea supporters to attend.
"Executives at Middlesbrough had been kind enough to extend their deadline for ticket sales and stadium allocation from 7.30pm last night until 9.30am this morning.
"It is important for the competition that the match against Middlesbrough goes ahead, however it is with extreme reluctance that we are asking the FA board to direct that the game be played behind closed doors for matters of sporting integrity.
"Chelsea recognise that such an outcome would have a huge impact on Middlesbrough and its supporters, as well as our own fans who have already bought the limited number of tickets that were sold before the licence was imposed, but we believe this is the fairest way of proceeding in the current circumstances.
"We will continue to discuss the issue of ticket sales with OFSI as there are a number of fixtures still to be played this season and we hope to reach a resolution."
