LONDON—Britain may seek to curb immigration from the European Union even before it formally leaves the bloc, the freshly-appointed government minister in charge of Brexit said Sunday.

David Davis, the secretary of state for exiting the European Union, said implementing a cutoff point for new arrivals from the EU might be necessary if numbers suddenly rise, swelled by foreign workers hoping to gain a settlement foothold before the laws change.

“One way of dealing with it could be saying OK, only people arriving before a certain date get this protection,” Mr. Davis said in an interview with Sky News broadcast on Sunday.

Mr. Davis, a Conservative Party veteran and longstanding euroskeptic, was on Wednesday appointed chief of the newly created government department tasked with overseeing Britain’s divorce from the EU.

Laying out his immediate priorities, Mr. Davis said securing the rights of EU citizens already residing in the U.K. was contingent on reaching a similarly “generous settlement for British citizens living in the EU.”

Thorny negotiations on the manner in which Britain ends its four decades of membership in the 28-nation bloc can only take place once the country formally invokes Article 50 of the Lisbon Treaty, the EU’s remaining members have said.

Once Prime Minister Theresa May gives official notice of the withdrawal, the U.K. will have two years to negotiate terms of separation. That includes the fate of the three million non-British EU citizens living in the U.K. and of Britons living elsewhere in the EU.

The starting pistol for Brexit should be fired by triggering Article 50 early next year, Mr. Davis has said.

Deciding when to formally invoke Article 50 has emerged as an early post-Brexit fault line between the U.K.’s major political parties.

On Sunday, Scotland’s leader, Nicola Sturgeon, said she would consider calling for a second referendum on Scottish independence if Ms. May were to push the Brexit button without Edinburgh’s buy-in.

In one of Ms. May’s first official acts, the prime minister met Ms. Sturgeon in Scotland on Friday.

Following their talks, Ms. May said she said she would hold off from triggering Article 50 until she had obtained U.K.-wide “approaches and objectives.”

Scotland has been a sensitive subject in the wake of the Brexit vote. The Scottish electorate voted overwhelmingly for Britain to stay in the EU, and the decision to leave has sparked enthusiasm among Scots for a referendum on whether Scotland should exit the U.K.

Ms. Sturgeon has said she is laying the groundwork for an independence vote. A 2014 referendum on Scotland’s independence from the U.K. failed.

Speaking on the British Broadcasting Corp’s Andrew Marr Show on Sunday, Ms. Sturgeon said she was ready to move on a second referendum if Scotland’s demands weren’t met, even if that meant doing so as early as 2017, if Ms. May decided to trigger Article 50 in December.

Brexit was also on the agenda for Britain’s opposition Labour Party, which is poised to carry out a leadership election after two lawmakers mounted a challenge against incumbent chief Jeremy Corbyn.

Owen Smith, a Welshman and former Corbyn ally who on Sunday became the second Labour lawmaker after Angela Eagle to take on Mr. Corbyn, said he was open to calling a second referendum on EU membership once it became clearer what a possible Brexit deal would look like.

“I think we need to make sure people are satisfied. We trusted people, rightly, to take the decision,” Mr. Smith said in an interview with the BBC’s Mr. Marr.

“We can trust them again in 18 months time to check it’s absolutely what they wanted,” he said.