Russian-Israeli billionaire Abramovich must not profit in the UK under the Government sanctions, and Chelsea are blocked from selling any more tickets in the terms of their new licence.
And after talks with the Government failed to yield a softening on that ticket stance, the Blues have now questioned the FA Cup quarter-final's "sporting integrity" should Chelsea fans be barred from attending.
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.
An earlier Chelsea statement read: "We are disappointed to announce we will not be able to sell tickets for Saturday's FA Cup tie at Middlesbrough.
"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 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.
"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."
A Football Association spokesperson confirmed that the FA will discuss the matter at the Professional Game Board meeting tomorrow. (Wednesday).