The police will "vigorously resist" the claim
Security firm Group 4's insurers have told the force they plan to sue under a Victorian law in which police can be held liable for damage during civil disturbances
The loss adjusters Capita McLaren have warned they will make a claim following last week's fire and riot.
A spokeswoman for Bedfordshire police authority said: "We find it quite outrageous and will resist it vigorously."
The claim is being made under the 1886 Riot Damages Act, which allows companies and individuals to sue the police over damage caused during civil disturbances.
A spokeswoman for Group 4 said the company had no involvement in the legal action.
Bedfordshire Chief Constable Paul Hancock said his first priority is to carry out a professional investigation into the riot on 14 February.
Police budget
He said earlier this week that he expected Group 4 to pick up the bill for policing for the riot.
About 25 detainees, mostly failed asylum seekers, are still on the run from Yarl's Wood.
Alistair Burt, the Conservative MP for North East Bedfordshire, said his constituents would be "astonished" by the move.
"Whoever is responsible for the fire and damage at Yarl's Wood, it certainly is not the police and they will be amazed that someone is trying to take £42m from the Bedfordshire police budget to pay for what happened.
"If this is the law of insurance, then I am a banana," he said.
Adrian Heffernan, the chairman of the Bedfordshire Police Authority, said that lawyers would be instructed to fight the claim.
Summer riots
Bedfordshire County Council leader Phillip Hendry said: "The bill is getting on towards the total bill for policing for Bedfordshire for the whole year.
"It's just an impossible situation," he said.
"We would obviously turn to the legal profession in the first instance and then may turn to the government."
Police authorities have been trying to get the government to repeal the section of the 1886 act after disturbances in the north of England last summer.
They say the bill is archaic and can no longer be justified.
The act was designed to charge the local police for failing to prevent people behaving in a "riotous or tumultuous" manner.
It was also invoked after riots in Moss Side, Toxteth, London and Bristol in 1981.