"After discussions with various parties, we have taken the decision to formally postpone the bout," promoters Matchroom and Wasserman Boxing said in a statement.
They said the BBBofC decision was "procedurally flawed and without due process".
"That remains a legal issue between the promoters and the Board which we intend to pursue."
Plans were thrown into doubt on Wednesday after the failed test emerged and the BBBofC prohibited the much-hyped fight between the sons of famous boxing fathers as "not in the interests of boxing".
The possibility of proceeding without the involvement of the BBBofC emerged on Wednesday as lawyers sought a solution but Benn's promoter Eddie Hearn ruled that out.
"This is not a situation where we want to run rogue and come up with an alternative plan," he told the BBC.
Matchroom said that "whilst there are legal routes to facilitate the fight taking place as planned, we do not believe that it is in the fighters' interests for those to be pursued at such a late stage, or in the wider interests of the sport."
The promoters called for a full investigation into what happened.
Benn returned a finding for "trace amounts of a fertility drug" in a random test carried out by the Voluntary Anti-Doping Association (VADA), reportedly in August.
He had subsequently passed a test carried out by the UK Anti-Doping Agency UKAD, the official testers for the event, and was not suspended by the BBBofC.
The fight between welterweight Benn and super-middleweight Eubank was set at a catchweight of 157 pounds.
"He's hugely disappointed. He was ready, he was on weight and he was very much looking forward to this weekend," Eubank's promoter Kalle Sauerland told a short news conference that took place without questions being taken.
The fight has been heavily promoted as a re-run of an epic clash 30 years ago between the boxers' fathers, Chris Eubank Sr and Nigel Benn.
