Candidates vying to replace David Cameron must guarantee that European Union nationals are not deported when Britain quits the bloc, campaigners from both sides have urged.
The group of campaigners calls for all leadership contenders to make a "clear and unequivocal" statement reassuring the three million citizens who are facing anxiety over their future, as well as a pledge to fight for a similar deal for UK citizens living in the other 27 EU countries.
Polling has found that 84% of the public supports allowing migrants to stay, including 77% of Leave voters and 85% of Conservative supporters.
Just 16% of voters want EU citizens to be forced to leave after Brexit, the ICM research for the independent think-tank British Future found.
Sunder Katwala, director of the think-tank, said: "What's very clear is that people did not vote to deport three million people living alongside them as their workmates and neighbours.
"The Prime Minister's statement that there would be 'no immediate changes' to the status of EU migrants offered more doubt than reassurance.
"Both he and those competing to succeed him now have a responsibility to make clear that EU migrants are welcome here and that any post-Brexit changes would apply only to new migrants."
A letter to the Sunday Telegraph calling for the commitment to be made has been signed by a range of public figures, including Brexit backers Daniel Hannan, a Conservative MEP, Labour's Gisela Stuart and Ukip's Douglas Carswell as well as Remain supporters Yvette Cooper, a Labour former frontbencher, and TUC general secretary Frances O'Grady.
Ms Cooper has written to Mr Cameron calling for him to take immediate action.
"The Prime Minister must not wait to sort this as it is causing great anxiety for EU citizens living here and British citizens living abroad," she said.
"And it is being exploited by extremists to run horrible 'Go Home' campaigns that no one wants to see in Britain."
Simon Walker, director general at the Institute of Directors, said: "What the country - individuals, businesses and families - need most urgently is a guarantee of stability.
"Members of the Institute of Directors are very concerned about what the UK's vote to leave means for their existing workforce.
"EU workers need reassurance that they can continue to work and live in this country, which is their home."
It comes as separate research found immigration fears that fuelled the vote for Britain to quit were driven by concerns that public services are being put under strain by migrants.
Nearly three quarters of voters, 71%, believe the pressure schools face from migrants is greater than any benefits that may be gained through their tax and staffing contributions, according to the 2015 British Social Attitude survey.
More than six in 10, 63%, also think the National Health Service is being stretched by immigration, according to the study by NatCen, an independent social research agency.
But just 35% believe immigration is bad for Britain's economy, down from 47% in 2013, while 40% think it undermines the country's cultural life, down 5% on three years ago.
The next prime minister will face intense pressure to curb immigration as the UK severs its ties with Brussels, but EU leaders have warned they will not compromise on freedom of movement if the country wants continued access to the single market.
Prof John Curtice, senior research fellow at NatCen, said: "This survey highlights that it is the belief that immigration puts pressure on public services that now appears to be the driving force behind public concern about the level of immigration into Britain."
:: ICM surveyed 2,418 GB adults online from June 24 to 26. NatCen carried out 4,328 interviews between July 4 and November 2 on a representative, random sample of adults in Britain with a response rate of 51%.