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's 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 Boro statement read: "We are aware of Chelsea's request to have Saturday's Emirates FA Cup sixth round tie played behind closed doors and find their suggestion both bizarre and without any merit whatsoever.
"All concerned are well aware of the reasons Chelsea have been sanctioned and that this has nothing to do with Middlesbrough Football Club.
"To suggest as result that MFC and our fans should be penalised is not only grossly unfair but without any foundation.
"Given the reasons for these sanctions, for Chelsea to seek to invoke sporting 'integrity' as reason for the game being played behind closed doors is ironic in the extreme.
"We currently await formal notification from the FA of the next steps but rest assured MFC will resist Chelsea's actions in the strongest terms."
Middlesbrough have knocked out Manchester United and Tottenham Hotspurs from this season's FA Cup.
