LONDON — Theresa May risks being held hostage by her own voters in Brexit negotiations, according to new polling data that shows Conservative supporters oppose any immigration system that does not lead to a sustained fall in the number of migrants coming to the U.K.

The prime minister and her Brexit secretary, David Davis, have signaled a softer stance on immigration in recent days, with May refusing to rule out a temporary continuation of free movement during a transition period after Brexit, and Davis saying that immigration levels would go up and down as required when Britain leaves the EU.

Foreign Secretary Boris Johnson said Thursday it was possible that free movement could continue during an implementation phase after Brexit.

However, new polling data from Greenberg Quinlan Rosner Research (GQRR), shared exclusively with POLITICO, reveals that a majority of Conservative voters want to see migration fall rather than fluctuate regardless of any mitigating factors — such as a guarantee that migrants have a job in Britain before they arrive — that might form part of a post-Brexit visa arrangement for EU citizens.

May has pledged to take the U.K. out of the single market, ending freedom of movement and restoring U.K. control of immigration from Europe.

Like her predecessor, David Cameron, she has a target of bringing immigration down to “sustainable levels,” with net migration slashed from its current level of around 300,000 a year to less than 100,000.

The policy is popular. Overall, 58 percent of the people surveyed agreed it is “essential” for the immigration system to bring down the number of migrants, against only 18 percent who disagreed. Among Conservative voters, 76 percent said immigration levels must fall.

But May is under pressure from businesses and many MPs who fear that a drastic reduction would harm the economy and create a skills shortage. The Open Britain group, an offshoot of the Remain campaign, this week called on the government to drop its immigration target, with support from the Chartered Institute of Personnel and Development and former Tory business minister Anna Soubry, the Independent reported.

The U.K.’s post-Brexit immigration system will not be determined directly by talks with Brussels, but drawn up by officials in Whitehall. However, leaders may have to negotiate whether the U.K. allows free movement to continue temporarily as a quid pro quo for single market access during a transitional phase leading up to a free trade deal with the EU.

Some EU countries could also demand generous visa access for their citizens as part of any EU/U.K. free trade deal, as some Eastern European countries did during the EU-Canada trade talks.

The prime minister will have to balance the needs of the economy with the demands of Tory voters.

The government is said to favor a work permit system, while many Brexiteers MPs also support curbs on welfare payments for migrants. However, neither a system that ensured migrants “have a job and skills that we need,” nor one that required migrants to pay taxes for several years before receiving benefits, would reconcile a majority of Tory supporters to the idea, suggested by Davis, that migration levels would continue to rise, according to the GQRR.

Among Conservative voters, 55 percent want to see immigration levels continue to fall rather than fluctuate, even if migrants have a job or skills, and 62 percent want to see levels continue to fall even if migrants don’t receive benefits for several years.

GQRR, which conducted polling for the Labour party until 2015, surveyed nearly 2,000 voters between March 8-10, with half of that sample asked to respond to questions on immigration policy.

Peter McLeod, vice president of GQRR, said May’s Conservatives could struggle to find an immigration policy that managed the needs of the economy without driving voters into the hands of UKIP.

“Brexit is a huge test for politicians in so many ways, but managing people’s expectations on immigration will be one of the toughest parts of it,” he told POLITICO.

“While Labour could plausibly go into an election with a flexible migration policy that keeps most of its voters happy, it’s still a nightmare for the Tories. As far as we can see from this poll, anything that doesn’t make a serious dent in migrant numbers will alienate a majority of Tory voters, and of course UKIP will be there to pick them up.

"The question for Theresa May and her cabinet is whether they can tolerate seriously damaging the economy and annoying the many U.K. industries that rely on EU workers in exchange for an easier ride from voters on this one issue.”

Labour voters take a softer stance on immigration, according to the poll.

While 48 percent think it is “essential” that migrant numbers fall from current levels, a majority — 51 percent — would be content to see immigration levels fluctuate if migrants had a job to come to or skills Britain needs.

A majority of Labour voters — 53 percent — would also support immigration continuing to rise and fall, so long as employers were not allowed to undercut the wages of British workers; a policy goal that has been prioritized by Labour leader Jeremy Corbyn.

This article has been updated to correct figures in the 'Jobs and skills' graphic.