Under the proposed terms, the bloc would still conduct customs checks on some goods flowing from Britain to Northern Ireland if those goods were ultimately destined for the European Union.
There would be a complex series of rules on tariffs and value-added tax payments to compensate for differences in tariff rates between the European and British customs unions, though negotiators were struggling late on Tuesday to resolve the issue of how to rebate value-added tax payments.
The arrangement would also be subject to consent by the Northern Ireland Assembly, but in a way that would prevent the Democratic Unionists, who have opposed previous such proposals, from simply vetoing it at the first possible opportunity.
The Democratic Unionists are crucial to Mr. Johnson’s effort to win a majority for the deal in Parliament. Their opposition to similar previous versions of a Brexit agreement forced Mr. Johnson’s predecessor, Theresa May, to overhaul that agreement to place all of Britain in the European customs union for a period of time.
Mrs. May’s deal was nevertheless soundly defeated in Parliament three times.
Mr. Johnson was seen as having a better chance of cobbling together a majority, in part because he was a vocal supporter of Brexit before the 2016 referendum and thus has greater credibility with euroskeptic elements of the Conservative coalition.
As British negotiators were huddling with their European counterparts in Brussels, Mr. Johnson met with a parade of skeptics in 10 Downing Street. His hope is to win approval of the deal in Brussels by Friday, and then put it to a vote in a special session of the House of Commons on Saturday.
Mr. Johnson has vowed to leave the European Union, with or without a deal, by Oct. 31, and his negotiators have labored to seal an agreement by this week so that the prime minister is not force to ask Brussels for an extension.