The First Minister is to meet with representatives of EU member states in Edinburgh on Tuesday
Nicola Sturgeon has written to David Cameron and the five candidates bidding to replace him, saying it was "imperative" to "respect" the rights of Scotland's 173,000 European Union citizens.
The First Minister's demands for "immediate guarantees" on EU nationals comes ahead of a meeting with consuls-general of EU member states and diplomats at her official residence in Edinburgh on Tuesday.
The summit follows a series of talks in Brussels with European political leaders aimed at securing Scotland's place in the EU.
Scots voted to remain in the EU by 62% in the referendum, but 52% of British voters as a whole voted to leave.
Ms Sturgeon said: "Scotland voted overwhelmingly to stay in Europe, yet citizens of EU countries who live, work and contribute to our country are understandably anxious and uncertain about what the UK referendum result means for them and their families.
"People from EU countries are an important part of Scotland's future. I am therefore seeking immediate guarantees from the Prime Minister, and all Conservative leadership candidates, that the residency status and the other existing rights of the 173,000 EU nationals living in Scotland will remain unchanged, now or in the future.
"This is a commitment that can and should be made and enforced now."
In a similar move, 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 - have all signed a letter to the Sunday Telegraph calling for Mr Cameron to take immediate action.
Ms Cooper said: "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.
"And it is being exploited by extremists to run horrible 'Go Home' campaigns that no one wants to see in Britain."
Last week, the British prime minister said European citizens living in the UK would face no immediate changes in their circumstances.
Speaking in the House of Commons, Mr Cameron said: "Leave campaigners were fairly clear that they wanted to protect the rights of people who are already here who have come to live, work and study, but obviously the final clarification of that and of the rights of British people living in other parts of the European Union will have to wait for the complex negotiations."
Meanwhile, an ICM poll for the independent think-tank British Future 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 research showed.
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.
63% also think the NHS is being stretched by immigration, according to the study by NatCen, an independent social research agency.