The Brexit minister has warned that EU citizens may be blocked from staying in Britain permanently even if they arrive before the country leaves the union.
David Davis said a "generous settlement" would be negotiated for EU migrants living in the UK and British citizens living in Europe.
But he warned this could attract a surge of EU citizens moving to Britain "in a big rush to try to grab a set of advantages that we are putting in place for people who have come here expecting us to remain with the European Union forever".
As a result, he said those who arrive after a set date yet to be decided could be blocked from being given an indefinite right to stay in Britain, he said.
Mr Davis also reasserted his belief that the EU would grant Britain access to the single market as well as a suspension of free movement rules, something which European leaders have so far ruled out.
His comments come as he also insisted Scotland cannot have a veto over any deal to leave the EU, despite Theresa May suggesting all of the UK should agree a unified approach.
Mr Davis dismissed suggestions made by Nicola Sturgeon that Scotland could stay in the EU while the rest of the UK leaves.
Speaking on Sky's Murnaghan show, Mr Davis said the UK will be left with a land border with the EU in the Republic of Ireland that will create difficult issues, and it would be a mistake to create another border in the north too.
He said: "I don't think that works. One of our really challenging issues to deal with will be the internal border we have with southern Ireland, and we are not going to go about creating other internal borders inside the United Kingdom.
"The aim here is to try to address the concerns of people who are basically Remain people, who say 'well we are worried about inward investment, we are worried about trade with Europe, we are worried about all sorts of things'.
"And we will try as best we can - they can't have a veto because there are 17.5 million people who have given us a mandate, they have told us what to do, we can't disobey it - but what we can do is to try to do what we can to minimise any disruption or turbulence or problems."
He also suggested that Article 50, the legal process through which the UK would officially set the clock ticking on its two-year Brexit negotiations, would be triggered "early next year".
His remarks paint a different picture of Brexit than that presented by the new Prime Minister, who has said she will try to build consensus before launching formal negotiations.
On Friday, Mrs May told Ms Sturgeon she would not trigger Article 50 before getting a UK-wide agreement.
Mr Davis, who is part of a triumvirate of Brext-backing new Cabinet ministers alongside Foreign Secretary Boris Johnson and the International Trade Secretary Liam Fox, has admitted that "even within government there'll be tensions" over the exit strategy.
Earlier, Australia called for a free trade deal with Britain as soon as possible in a Brexit boost for the Government.
In a Saturday phone call, Mrs May spoke to her Australian counterpart Malcolm Turnbull, who expressed his desire to open up trading between the two Commonwealth countries as a matter of urgency.
Mrs May said: "I have been very clear that this Government will make a success of our exit from the European Union.
"One of the ways we will do this is by embracing the opportunities to strike free trade deals with our partners across the globe.
"It is very encouraging that one of our closest international partners is already seeking to establish just such a deal.
"This shows that we can make Brexit work for Britain, and the new Secretary of State for International Trade will be taking this forward in the weeks and months ahead."
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