The Premier League club want Saturday's game at the Riverside played in an empty stadium after the British government blocked them from making new ticket sales, with only existing ticket holders allowed to attend matches.
The Blues have pushed ministers for relaxations to the strict operating licence put in place after Russian owner Roman Abramovich was sanctioned last week as part of a set of punitive measures following Russia's invasion of Ukraine.
The licence is designed to prevent Chelsea and by extension their billionaire owner -- described by the government as part of Russian President Vladimir Putin's inner circle -- from generating new revenue.
The club issued a statement on Tuesday calling for the FA Cup sixth-round tie to be played behind closed doors in the interests of fairness.
"It is with extreme reluctance that we are asking the FA (Football Association) board to direct that the game be played behind closed doors for matters of sporting integrity," Chelsea said.
"Chelsea FC recognises 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."
But the Championship club hit back with a strong statement of their own, making a pointed reference to the reasons Chelsea were in their current situation and saying they would resist the club's actions "in the strongest terms".
"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," Middlesbrough said in a statement.
"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."
The FA said the matter would be discussed on Wednesday.
“We are aware of Chelsea FC’s request to play their Emirates FA Cup quarter-final tie against Middlesbrough FC behind closed doors," said an FA spokesman. "The matter will be discussed at the Professional Game Board meeting tomorrow.”
