After meeting with Cypriot President Nicos Anastasiades, Barclay was asked if Britain would be leaving the EU on Oct. 31 without an agreement, and Barclay said "yes, we will."
Barkley said that he was "grateful to the president for the very strong bilateral relationship that we have."
"He has a deep understanding of the issues around Brexit and very much shares the prime minister's desire for a deal ... It was surely appreciated to get the president's engagement on this issue because both sides want to get a deal, which is in the interest of both sides, and that is what we are working for," Barclay said.
Barkley is on his visit to Cyprus to discuss Brexit issues. A spokesperson for the Department for Exiting the EU in London told the Cyprus News Agency that the visit was part of the British government's engagement with EU member states, "as the UK seeks to reach a deal with the backstop removed."
Britain has two bases on the island, which will have the status of foreign territory with boundaries with Cypriot soil.
However, the two countries have reached a private agreement that the status of the base areas will not change and that Cypriots living or working within the boundaries of the bases will not be adversely affected in their everyday life after a hard Brexit.
Likewise, Britain guaranteed that Cypriots living in Britain will not be affected and Cyprus guaranteed that about 80,000 Britons living on the island will not have their everyday life disrupted.
"We have made an unequivocal guarantee to the 30,000 Cypriot nationals living and working in the UK. They will have absolute legal certainty of the rights to work, study and access benefits and services on the same basis as now under any Brexit scenario," the Brexit ministry spokesman said.
He added that Barclay will be stepping up his engagement with Michel Barnier, the EU's chief Brexit negotiator, over the coming days, following their meeting in Luxembourg on Monday, which he described as constructive.