BBC Ban At Bolton
He is also livid with Radio Five Live pundit Alan Green for pointing the finger at Wanderers - during a show presented from the Reebok's Press room.
The Beeb were ordered out of Allardyce's Press conference yesterday and the Bolton boss also pulled out of a show he was due to appear on with referees' boss Keith Hackett.
He rapped: "We feel we were the innocent party in the Liverpool game and anyone with even a slight knowledge of football should be able to see that.
"We didn't kick anybody, stamp on anybody or threaten anybody's career with terrible tackles. We also didn't get a penalty we deserved or a couple of offside decisions.
"The things that were said after that game caused outcry in Bolton - but I understand the person responsible is a Liverpool fan anyway.
"The remarks he made don't deserve any other comment from me. I wouldn't want to contribute towards the BBC keeping him in a job just because he's prepared to say outrageous stuff."
Allardyce reckons Gerrard stamped on Bolton skipper Kevin Nolan, while Sissoko did the same to El-Hadji Diouf.
Neither player was booked - but Allardyce suggested referee Mark Clattenburg had a poor game because it was his fourth game in eight days.
Allardyce added: "Managers have done their best to rotate players during the holiday period but there was no respite for referees.
"And I don't think he would have made the same mistakes if this had been his first game instead of his fourth."
Meanwhile, Bolton Wanderers manager Sam Allardyce has told defender Martin Djetou to prove he's worth a new deal.
Allardyce said: "We will get through the Watford game and make a decision after that.
"One would hope Martin would go out and prove what he is worth. Then we would say 'yes'.
"I like the man immensely as a person but you have to be a good footballer.
"He has to do a real job, so I have to make that assessment in the next few days."