Sir Keir Starmer has said the UK’s benefits system is broken and fixing it is a “moral imperative”, a day after a backbench Labour revolt saw him forced into a U-turn on welfare cuts.

The prime minister told the Welsh Labour Party conference in Llandudno that the government would not take away the welfare “safety net that vulnerable people rely on”.

But he said he could not let benefits “become a snare for those who can and want to work”.

Despite the government’s concession on its plans to reform welfare, some Labour MPs want further changes, while the Unite union has called for the proposal to be dropped altogether ahead of a vote on Tuesday.

The BBC understands whips and cabinet ministers – including Wes Streeting, Angela Rayner and Rachel Reeves – have been phoning or texting Labour MPs over the weekend, going through the names of the initial rebels in a bid to get an accurate assessment of potential voting.

Some MPs are saying they have yet to make their mind up on how to vote and are awaiting a statement on Monday from Work and Pensions Secretary Liz Kendall that will spell out government concessions.

Speaking at the conference in north Wales on Saturday, Sir Keir said fixing the “broken” benefits system needed to be done because it was “failing people every day”, leaving “a generation of young people written off for good and the cost spiralling out of control”.

“Fixing it is a moral imperative, but we need to do it in a Labour way,” he added.

The government’s initial plans, aimed at bringing down the welfare bill, would have made it harder for people to claim personal independence payment (Pip), a benefit paid to 3.7 million people with long-term physical or mental health conditions.

But following a rebellion among Labour MPs and the likelihood the government would be defeated in the Commons, the government announced the stricter criteria would only apply to new claimants.

It reversed its plans to freeze the health-related component of universal credit, and the payment will now rise in line with inflation for existing recipients.

Ministers will also carry out a review of the Pip assessment process, with input from disability organisations.

A £1bn support package to help people into work, originally scheduled for 2029, will be fast-tracked.

A new “reasoned amendment” to the bill will be put down on Monday by rebel MPs, which will reflect government concessions but is expected to be similar to the now-withdrawn earlier amendment that sought to block changes to the benefits system.

The BBC understands that around 50 Labour MPs currently back that new amendment.

That number is likely to increase but the expectation is it will not reach the 80-plus needed to put the government in danger of defeat. However it would still represent a significant rebellion.

Rebel MPs are also expected to hold a briefing on Monday night at Westminster with various disability charities.

Labour MP Diane Abbott earlier told BBC Radio 4’s Today programme that she thought the result of a vote on the new plans would be tight, partly because backbenchers are still “upset about the lack of consultation” and because of “the notion of a two-tier benefit system”.

But former Labour justice secretary Lord Falconer told the programme that “sensible” changes to the welfare reforms were “pretty significant”, and that he believed opposition among Labour MPs was “shrinking and shrinking”.

Debbie Abrahams, the Labour MP who chairs the Work and Pensions Select Committee, told the BBC on Friday: “The concessions are a good start, they are very good concessions and they will protect existing claimants.

“However there are still concerns about new claimants. It would not be right for me not to do anything just to spare the prime minister an inconvenience.”

Ahead of Sir Keir’s conference speech, Unite called for the “entire welfare bill to be dropped and for the government to start again”, with general secretary Sharon Graham accusing Labour of “attacking the most vulnerable in our society”.

“The government’s latest plans for disabled benefits cuts are divisive and sinister,” she said.

“Creating a two-tier system where younger disabled people and those who become disabled in the future will be disadvantaged and denied access to work and education, is morally wrong.”