Ministers have been accused of repeatedly misleading MPs about the impact of their £26,000 cap on welfare payments after it emerged that Eric Pickles, the communities secretary, secretly warned the plan would cost more money than it saved and increase homelessness by 20,000.

Liam Byrne, the shadow work and pensions secretary, insisted that Pickles's comments, set out in a letter from his private secretary to No 10 that was leaked to the Observer, showed that a succession of ministers "haven't been straight with the House of Commons". They have either dismissed claims that the cap would increase homelessness, or insisted its likely impact was impossible to quantify.

The benefit cap, announced by George Osborne, the chancellor, to the delight of the Tory right at the Conservative party conference last autumn, is one of the most high-profile and controversial of the government's myriad welfare reforms. The welfare bill still has to go through the Lords and Pickles's letter will embolden peers seeking to amend it so the cap is less punitive.

The letter, sent on Pickles's behalf by Nico Heslop, his private secretary, explicitly says welfare cuts could make 40,000 families homeless. "Our modelling indicates that we could see an additional 20,000 homelessness acceptances as a result of the total benefit cap. This on top of the 20,000 additional acceptances already anticipated as a result of other changes to housing benefit," Heslop wrote.

The letter was sent in January. Since then, ministers and officials have made a series of Commons statements that Labour believes are hard to square with what Pickles was telling No 10 in private. Those highlighted by Labour include:

• The Department for Work and Pensions (DWP) publishing an impact assessment in February saying that it was "not possible to quantify" the cost to local councils generated by the welfare cap and the likelihood that it will require councils to house some families made homeless.

• Grant Shapps, the housing minister, citing the DWP's impact assessment when specifically asked by a Labour MP if he had an estimate of the number of households that would be made homeless as a result of the benefit cap.

• Maria Miller, a welfare minister, telling Karen Buck, a Labour MP, to "get real" when asked about the impact of the benefit cap on homelessness. "I do not accept that the policies we are advocating will have the impact on homelessness that she talked about," Miller said.

• Chris Grayling, another welfare minister, saying: "I do not deny that the benefit cap may result in individual cases of housing mobility [ie, people having to move], but I do not believe that the measure will exacerbate [the problem]."

Byrne said on Sunday night: "The idea that you can go out and say that there is no further evidence that you are aware of, four months after the Department for Communities wrote to the prime minister saying there was different evidence, is breathtaking.

"We want answers from Iain Duncan Smith [the work and pensions secretary] in the House of Commons about why his department hid official government evidence that his policy would make 40,000 families homeless."

Byrne's colleague Caroline Flint, the shadow communities secretary, said: "It has become clear that while Eric Pickles defends his government housing policies in public, in truth he doesn't believe in them. The public and parliament have a right to know why time and again his department dismissed the very same housing concerns he secretly raised with the prime minister."

It is understood that Labour will try to force Pickles and Duncan Smith to respond to an urgent question on this in the Commons chamber on Monday. But it is up to the Speaker, John Bercow, to decide whether to accept the move.

In the letter, the Department for Communities and Local Government suggested that the impact of the policy could by ameliorated by ensuring child benefit is not included in those benefits that count towards the cap. But on Sunday , the DWP, which is in charge of the plan to impose a £26,000 cap on the total amount of benefits than can be claimed in any year by an unemployed family, confirmed that Pickles's proposal had been rejected and that child benefit would be taken into account when the cap comes into force in 2013.

In the letter, Heslop also claimed the benefit cap would cost the exchequer money. Although it was projected to save £270m, that sum "does not take account of the additional costs to local authorities (through homelessness and temporary accommodation)," he said. "In fact, we think it is likely that the policy as it stands will generate a net cost."

He also said that up to 23,000 affordable rental units could be lost because the benefit cap would stop developers charging the rents they wanted, giving them less incentive to build property.

The DWP said it did not recognise the figures in the letter, and did not accept the cap would increase homelessness.

"You know what councils are like – when they have concerns, they are very vocal about it," one source said.

"The cap only comes in at £26,000 and that's equivalent to a gross income of £35,000 for a family that's working. And the minute someone enters into part-time work, they are exempted from the cap," the source went on. "There might be some people who have to move to a less expensive area. But that doesn't mean they won't have anywhere to live. We are very optimistic about the behavioural change that this will bring about. We have already started to change housing benefit. And have you seen droves of homeless people? No, you have not."

The Department for Communities said Pickles was "fully supportive of the government's policies on benefits". A source said Pickles had not personally raised the issues set out in the letter with cabinet colleagues, either formally or informally.

A spokesperson for the DWP said: We cannot carry on with a situation where people on benefits can receive more in welfare payments than hard-working families and where a life on benefits robs people of achieving their potential. No one needs to be homeless because of these reforms. Many working families live on this amount of money."