Following the freezing of Russian owner Roman Abramovich's assets, the government granted Chelsea a license to continue playing.
Chelsea is not permitted to sell tickets to any of their matches under the rules of the agreement.
Middlesbrough has stated that they will fight the "bizarre" and "ironic" request "to the fullest extent feasible."
Middlesbrough said that Chelsea's request, which was made "on matters of sporting integrity," was "without any validity whatsoever."
"Chelsea and sports ethics do not belong in the same sentence," Middlesbrough chairman Steve Gibson told The Athletic.
"Where is the intelligence of Bruce Buck, Chelsea's chairman, who has been an apologist for his owner, where the trophies won over the past 19 years have come from Abramovich's corrupt money?" Gibson went on to say.
"Where is the intellect of the chairman of Chelsea when it comes to playing his games at home in front of his season-ticket holders? Does he want to play all his away games at empty stadiums?
"Abramovich has been kicked out of football and Bruce Buck should be kicked out of football too."
The Football Association will make a decision on Wednesday.
Chelsea Supporters Trust asked the club to withdraw the request, saying it "does not benefit" any fans.
Chelsea had sold between 500 and 600 tickets out of a total of 4,620 before the club's license restrictions took effect on March 10th.
The safeguard was put in place to prevent oligarch Roman Abramovich from benefitting from the sale of Chelsea.
After the UK government sanctioned him for his ties to Russian President Vladimir Putin, he was barred from becoming a club director and the sale of Chelsea was postponed.
Chelsea claimed they asked the FA "with considerable reluctance" not to allow fans of any team to attend the game at the Riverside "for reasons of sports integrity."
