Iain Duncan Smith has insisted his welfare reform plans are not designed to "punish", as he vowed the proposed cap on benefits will not make children homeless or force families into poverty.

The work and pensions secretary, who is facing a threat from Lords to vote-down the government's proposed £26,000 per year benefit cap, gave a guarantee the reforms would not leave children on the street.

"The reality here is that £26,000 a year, there is absolutely no reason why any family should not be able to be found accommodation and no children should be in any respect plunged into poverty," he told BBC Radio 4's Today programme.

Impact assessments show the government expects 67,000 households - and 220,000 children - will be affected by the changes.

"We do not believe there will be an increase in child poverty," Duncan Smith said. "Our department does not believe that you can directly apportion poverty to this particular measure."

Prime Minister David Cameron said: "It's a basic issue of fairness.

"Should people really be able to earn more than £26,000 just through benefits alone?

"I don't believe they should. And I think the overwhelming majority of people in the country would back that view." .

Speaking in Leeds Mr Cameron said the £35,000 equivalent wage was a "good, healthy salary".

The prime minister's official spokesperson said on Monday the government would look at the implementation. "We will look at the impact of these changes and see if there is anything more we have to do on implementation.

"But we are sticking to the fundamental policy. We think it is fair to have a cap and we think that most people would think that a cap which equates to £35,000 gross income is quite a lot of money."

Responding to alliance of peers and Bishops, including former Liberal Democrat leader Lord Ashdown who are planning to derail the reform, Duncan Smith said the cap was not designed to "punish" and his critics were wrong.

"I simply make the point to them that the purpose of this is not to punish people but it is to give fairness to people who are paying tax, who are commuting large distances because they can only afford to live in the houses that they have chosen," he told Sky News.

"It is also about fairness to those who are on these benefits; it is not fair to trap somebody in an expensive house which they cannot afford then to go to work on the back of, because they would lose their housing benefit if they went to work - so they are disincentivised from going to work.

The cap, which would mean no family would be able to claim over £500 a week in benefits, was denounced as "completely unacceptable" by Lord Ashdown on Sunday.

Despite the divisions in his own ranks, however, Deputy Prime Minister Nick Clegg has said he is fully signed up to the changes.

Clegg suggested there was some scope for softening the impact of the changes through "transitional arrangements" around the introduction of the cap.

However, he flatly rejected an amendment tabled by the Bishop of Ripon and Leeds, the Rt Rev John Packer, which would exclude child benefit payments from the £500-a-week limit.

Shadow work and pensions secretary Liam Byrne has said Labour will support the bishop's amendment to exclude child benefit from the cap if the party's own amendment is defeated.

Employment minister Chris Grayling said some families would have to move house because of the changes: "There certainly will be people who have to move house as a result of this, who have to move to a part of town they can afford to live in, but surely that is right," he told BBC Radio 5 Live's Pienaar's Politics.

On Sunday Iain Duncan Smith hit out at bishops trying to block his welfare reforms, accusing them of ignoring the concerns of ordinary people.

In an interview with The Sunday Times, he acknowledged that his plans to limit the total payments any household can receive could face defeat in the House of Lords on Monday.

He urged the bishops, who are leading the opposition in the upper chamber, to rethink their objections, insisting they were not doing the poor any favours.

Labour have said they will not vote against the bill "because we support the principles" but would seek to change it: "We will be seeking to amend the bill, to bring a compromise between the bishops and the Government because we don't think council taxpayers should be hit with a massive bill for homelessness," a party spokesperson said.