Workers will feel "betrayed" if Britain does not control its borders after quitting the EU, the challenger to Unite's Len McCluskey will warn.
Gerard Coyne said the government should "not even begin to negotiate" over immigration as it attempts to look for common ground in a Brexit settlement.
The warning from the trade union leadership candidate, vying to oust McCluskey, one of Jeremy Corbyn's close allies, comes as Labour divisions grow over freedom of movement.
Mr Coyne, currently Unite's West Midlands regional secretary, will say that migration within the EU has benefited the better-off by allowing them to hire cleaners and nannies, but has put pressure on services for others.
Prime Minister Theresa May must now say that curbing freedom of movement is non-negotiable, and that will lead to the UK's exit from the single market, he will tell Unite members at a speech in Birmingham.
"The better off have been able to hire Europeans as their cleaners, or nannies, and have their cars washed at little cost, by people eager to work and prepared to accept what are, by UK standards, low wages," he will say.
"But for the many Britons facing insecurity in the job market, who rely on public services such as the NHS and state schools, and who need affordable homes, the presence of a very large number of foreign nationals has added to the pressures they already face at a time of austerity.
"Theresa May and other ministers should not wait until Article 50 has been triggered to set out a negotiating position on free movement of labour. They should be saying now, without equivocation, that the issue is non-negotiable.
"There can be no compromise on the principle of taking back control of our borders."
Labour's policy on immigration has been in disarray for months as senior party figures have been at odds over the issue. Shadow Brexit secretary Sir Keir Starmer reiterated his calls for curbs on freedom of movement, saying a "fundamental rethink of immigration rules" was needed, in an interview with the Sunday Times just a week after the Labour leader indicated he wanted the policy to continue post-Brexit.
"My many conversations with Unite members leave me in no doubt that those who voted for Brexit expect that promise of an end to uncontrolled immigration from the EU to be kept, and will feel betrayed if it is not," Mr Coyne will say.
"Let us not fool ourselves. Brexit means exit. It means a world in which we have to be competitive enough to thrive outside the single European market."
Mr McCluskey's five-year term as general secretary of the union was due to end in 2018 but he chose to bring the election forward to spring.