David Davis says new EU migrants to the UK may be forced to leave
In Britain, the newly-appointed Brexit minister says the border between Ireland and the UK will be one of the "really challenging issues" to deal with.
David Davis also says EU migrants who arrive between now and the UK's departure from the bloc may not be given the opportunity to stay, in order to prevent a last-minute rush to Britain.
Mr Davis wants to secure "a generous settlement for EU migrants here now and a generous settlement for British citizens in the EU" - but this may not apply to newcomers.
On the possibility of a surge of migrants before the departure, Mr Davis told the UK Mail On Sunday: "We may have to deal with that. There are a variety of possibilities."
"We may have to say that the right to indefinite leave to remain protection only applies before a certain date. But you have to make those judgments on reality, not speculation."
Mr Davis also claimed Article 50, the two-year mechanism for leaving the EU, should be invoked by early 2017 - meaning the UK would be out by 2019.
He also suggested Scotland will not have any veto over the decision.
Meanwhile, Australia has called for a free trade deal with the UK as soon as possible.
New British Prime Minister Theresa May spoke to Australian counterpart Malcolm Turnbull on Saturday.
She said the phone call proved leaving the European Union could work for the UK.
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."
A new poll suggests Mrs May should not call a second referendum on any deal the UK reaches with the bloc.
In a ComRes poll for the UK's Sunday Mirror and The Independent, 57% rejected calls for a second referendum - an idea backed by Labour leadership candidate Owen Smith.
Some 29% were in favour.