Jeremy Corbyn will say on Wednesday that a Labour government will not “sow division” by promising to cut immigration, despite pressure from key backbenchers to shift the party’s stance after the Brexit vote.

Freedom of movement has emerged as a fresh dividing line in the party, with some MPs saying the message from the public in the vote was that levels of immigration from the EU had become too high.

But the Labour leader will tell his party: “A Labour government will not offer false promises. We will not sow division or fan the flames of fear. We will instead tackle the real issues of immigration – and make the changes that are needed.”

His spokesman told journalists: “He is not concerned about numbers,” adding that rather than seeking controls on immigration, Labour would seek to mitigate its effects on low-paid workers by reintroducing a “migrant impact fund”.

A £50m fund to ease the pressure of migration on public services was introduced by Gordon Brown in 2008 and scrapped two years later by David Cameron under the coalition in summer 2010. “As long as the consequences of immigration are tackled, it is not an objective to reduce the numbers, to reduce immigration,” the spokesman said.

Corbyn repeatedly said during the referendum campaign that he did not believe Britain should seek to cut immigration. The prime minister, Theresa May, has made clear she will seek controls on freedom of movement when Brexit negotiations commence; and some Labour MPs believe that message will resonate with working-class voters who feel their wages are under pressure.

The Leeds West MP, Rachel Reeves, told a fringe meeting on Tuesday that her constituency was like a “tinder box” and voters’ concerns about immigration and lack of control over their lives could “explode” on to the streets if they remain unaddressed.

Reeves is among several prominent backbenchers, including Chuka Ummuna, Stephen Kinnock and Emma Reynolds, who have responded to the Brexit vote with calls to reopen the debate on freedom of movement.

The shadow home secretary, Andy Burnham, will strike a different tone from Corbyn in his conference speech, joining the calls for immigration reform. Burnham will call for a “fair Brexit” that addresses the concerns of people who “feel taken for granted and abandoned by the political class”.

“Labour must face up fully to this fact: millions of our lifelong supporters voted to leave the EU and voted for change on immigration,” he will say.

Reeves, a former shadow work and pensions secretary who sits on the Treasury select committee, told a fringe meeting: “There are bubbling tensions in this country that I just think could explode.” She said there had already been three racist incidents since the referendum and she feared there could be more cases if the public’s concerns are not addressed.

She told the Guardian: “This referendum has revealed that there are a lot of people [who] feel they don’t have a voice, that this country is not working for them and they have faced a combination of years of austerity and a rapid pace of change in their community they didn’t vote for.

“I’m saying we need to have some controls on immigration,” she said, adding, “You can’t just close down that discussion or label people as racist if they say that.”

Corbyn’s restatement of his commitment to uncontrolled immigration comes as he tries to tempt backbenchers to return to fill more than 60 posts in his shadow cabinet vacated by rebel MPs who resigned en masse after the referendum.

Talks between Corbyn and potential shadow ministers are under way and he said he plans to complete a reshuffle before parliament returns after the party conference season – which would mean announcing new recruits next week.

Several Labour sources told the Guardian that there is a “nascent” plan to bring about peace and make sure all wings of the party were represented at the top. One senior Labour MP said it could work in a similar way to the Australian Labor party’s caucus system, which formally splits into factions of which the “national left” and “national right” are the biggest.

A “soft left” centrist group is already emerging in the form of a revived Tribune alliance, which was a powerful influence within the Labour party in the years of Harold Wilson.

Another group calling themselves the “Clause I Socialists” on the right of the party is also developing, while there are dozens of MPs on the left of the party who are loyal to the leader.

Corbyn’s closest ally, the shadow chancellor, John McDonnell, said: “Look at Harold Wilson’s cabinets. When you construct a shadow cabinet and a cabinet, you choose from left, right and centre. They give you a hard time, there’s a real rumbustious debate and then you make a decision.” He added that having representatives of all wings of the party in the shadow cabinet was “a reflection of reality”.

One of the MPs who is considering returning to the fold said it would be necessary to have formal “terms and conditions”.

Clive Efford, who is convening the new Tribune group, said about 60 MPs had signed up so far but he believed it would soon reach between 80 and 100. MPs have to ask to be allowed to join and must share the values of the group. “The Tribune group is a vehicle to develop ideas and communicate them to party members, to stimulate debate and also to let them know that the parliamentary Labour party is trying to develop ideas,” Efford said. “It is very much a traditional left-of-centre group.”

Corbyn insists he can assemble a full team without having to allow MPs to elect their colleagues to the frontbench – a suggestion made by the deputy leader, Tom Watson, as a way of allowing rebel MPs to return with “dignity”.

However, Corbyn also appeared to undermine an intricate compromise announced by the shadow defence secretary, Clive Lewis, on the Trident nuclear deterrent, which Lewis and others had hoped would help persuade some disaffected MPs to return.

Lewis, who has taken over Labour’s defence review from the shadow foreign secretary, Emily Thornberry, suggested on Tuesday that Labour would not come back to the issue of Trident before the next general election – though Corbyn would be free to campaign on the issue.

Corbyn said: “The policy the party has from previous conference decisions does support the renewal of Trident. As you know, I never agreed with that decision. That’s the existing party policy. I cannot predict what will happen in the future, who will decide what they want to bring forward to conference.”

In his closing speech to conference on Tuesday, Corbyn will urge the Labour party to be ready for an early general election and urge members to reject the “trench warfare” that has characterised recent months.

Watson, who urged Corbyn to resign over the summer, received an ovation in the conference hall for his speech on Tuesday, urging his party to stop “trashing the record” of the Blair and Brown governments if it wants to win elections. “I don’t know why we’ve been focusing on what was wrong with the Blair and Brown governments for the last six years but trashing our record is not the way to enhance our brand,” he said. “We won’t win elections like that and we need to win elections,” he said.

Responding to the Labour leader’s statements on migration, the home secretary, Amber Rudd, said: “All this shows is that Jeremy Corbyn wants unlimited immigration – and presides over a Labour party that is too divided, incompetent and distracted to lead our country.”