Mr. Varadkar has been blunt about his objections to Mr. Johnson’s proposal, notably its requirement that Northern Ireland’s assembly — which has been suspended since its governing coalition collapsed in January 2017 in a sectarian dispute — be given a veto over the plan once every four years.

Before the meeting, Mr. Varadkar said, “Part of the difficulty at the moment, though, is the position of the U.K. government that Northern Ireland must leave the E.U. customs union and be part of the U.K. customs union, no matter what the people of Northern Ireland think. That’s their position at the moment, and that’s the one that’s of grave difficulty to us.”

British officials have floated various remedies to this problem, like having the major political parties in Northern Ireland — which represent both British unionists and Irish nationalists — sign on to an agreement, in lieu of the assembly. Another option is to impose a time limit on the deal, but set that date well in the future, giving Ireland hope that the north might never be divided from the south.

Finding a compromise on the issue of “consent” was expected to occupy most of Thursday’s meeting, though Mr. Johnson and Mr. Varadkar also said they had discussed the customs union. Of the two issues, consent is easier to resolve: Some of the details could be left until later, if the two sides reach agreement on basic principles.

The customs union is far harder: Britain argues it is necessary as a point of principle and to get a deal through Parliament. But it would require installing customs checks, although Britain insists such checks could be done away from the border between north and south.

Those are an obstacle for Ireland for practical reasons — they would interrupt the now seamless trade across the border — as well as a symbolic one — a hard border could threaten the peace that has prevailed since the 1998 Good Friday Agreement resolved years of sectarian strife in Northern Ireland.

The European Union and individual European leaders have steadfastly backed Ireland, generally echoing Mr. Varadkar’s objections. On Wednesday, Mr. Barnier told the European Parliament: “We’re not really in a position where we’re able to find an agreement.”