Ahead of the meeting in Brussels, Barclay on Thursday warned the EU against a "rigid" approach and suggested the final details of an alternative to the Irish backstop may not need to be resolved until the end of 2020, the BBC reported.
The meeting on Friday comes after European Commission President Jean-Claude Juncker said a new Brexit deal could still be reached before the October 31 deadline.
Juncker told Sky News on Thursday that he had no "emotional relationship" with the backstop the controversial policy aimed at preventing the return of a hard border on the island of Ireland after Brexit adding: "If the objectives are met - all of them - then we don't need the backstop."
He said his meeting with UK Prime Minister Boris Johnson in Luxembourg on Monday was "rather positive", and "we can have a deal" in the next few weeks.
Johnson, who has said he wants to leave the EU, preferably with a deal, on October 31, has urged the EU to scrap the backstop.
The EU has asked for alternative proposals and had previously criticised the UK for not putting any plans in writing.
On Thursday, the UK government said confidential documents that "reflect the ideas the UK has put forward" on Brexit had been shared with the EU.
It came after Finland's Prime Minister said Johnson had 12 days to set out his Brexit plans to the EU - although a government source said the development was not in response to those remarks.
Johnson will hold more talks with European leaders at a UN summit in New York next week.
Irish Prime Minister Leo Varadkar, who will be among those at the summit, said he would try to get a deal with Johnson, reports the BBC.
Thursday also saw the final day of the legal battle over Johnson's decision to suspend Parliament at the UK's Supreme Court. The President of the Supreme Court, Lady Hale, said the justices would announce a decision early next week.
The Prime Minister prorogued Parliament earlier this month for five weeks, with MPs not scheduled to return until October 14.
( With inputs from IANS )