LONDON — A small but influential party aligned with Prime Minister Theresa May’s government suggested on Friday that it still cannot support her plan for Britain’s departure from the European Union, a development that could doom her hopes of persuading Parliament to pass the deal next week on the third try.

The Democratic Unionist Party of Northern Ireland holds just 10 seats in the 650-seat House of Commons. But the central sticking point in the talks on a British withdrawal has been how to handle the border between Northern Ireland and Ireland, and the party’s position carries disproportionate weight with others in Parliament.

Parliament has twice rejected the agreement that Mrs. May negotiated with the European Union for Britain’s departure, known as Brexit, and both times the D.U.P. has voted against it, largely because of concerns about the border issue.

If Parliament rejects the deal again, there is still the possibility of a cliff-edge Brexit in the short term or a much longer delay to renegotiate the agreement to leave.