DUBLIN (NYTIMES) - There is a lurking suspicion among the Irish that Prime Minister Boris Johnson is the latest in a long line of British leaders who have not cared much about them.

He took nearly a week to return a congratulatory phone call from Ireland's prime minister Leo Varadkar, and he has brushed aside the challenge of enforcing a hard border between Northern Ireland and the south, once saying it would be no different from collecting traffic fees from motorists driving across London.

So, when Mr Johnson made his first visit to Dublin as prime minister earlier this week, he went out of his way to show he was sensitive to the threat that Britain's departure from the European Union poses to its smaller next-door neighbour.

Brexit, he said, was a "conundrum that Ireland never asked for." As Mr Johnson seeks to hammer out a new deal with Europe on Britain's departure, he is weighing a proposal that would put parts of the Northern Ireland economy into an "all-Ireland" zone, presumably subjecting them to EU rules and standards and thereby preserving the north's open border with the south.

For the record, Mr Johnson continues to rule out keeping all of Northern Ireland in the European Union's economic orbit - a step that is fiercely opposed by Northern Ireland's Democratic Unionist Party, Mr Johnson's coalition partner, because it views that as an unconstitutional severing of the north from Britain.

But people who have spoken to British officials say the concept is still alive, if in piecemeal form, with the government focused on food and agriculture; the cross-border trade in beef, milk, and other products is especially susceptible to disruption.

Mr Johnson may have few other options: Parliament has forbidden him to leave Europe without a deal, and it rebuffed his call for an election before the Brexit deadline of Oct 31. How he handles the bedeviling issue of the border could determine whether he achieves his goal of a swift exit from Europe - or even survives in office.

"All routes now lead to Ireland," said Mr Bronwen Maddox, director of the Institute for Government in London, who recently travelled through Northern Ireland. "It is the issue the world is interested in, precisely because so many countries poured their energies into getting peace in Ireland."

In Washington, House Speaker Nancy Pelosi warned that Democrats in Congress would hold up any trade deal Britain signs with President Donald Trump if Brexit jeopardises the Good Friday Agreement.

The 1998 pact, which created a power-sharing arrangement between unionists and nationalists in Northern Ireland, is a cherished legacy of President Bill Clinton, whose envoy, Mr George Mitchell, helped broker it.

Mr Johnson has promoted a trade deal with Washington as one of the incentives of leaving Europe; getting tied up in Congress over Ireland would be an embarrassing setback. There are signs that people on all sides of the debate are starting to recognise the trans-Atlantic repercussions of the issue.

For Britain, a new deal on Northern Ireland could be a remedy that would satisfy Ireland, the European Union and most of the British public. Analysts said Mr Johnson could risk cutting loose the Democratic Unionists. Having already lost his Parliamentary majority and called for an election, he does not face the same danger of a no-confidence vote that haunted his predecessor, Mrs Theresa May.

"It's the only game in town," said Mr Thomas Wright, director of the Centre on the United States and Europe at the Brookings Institution. "It may not happen, but it offers an off-ramp, and it's basically the only off-ramp being offered."

The idea of leaving Northern Ireland in the European customs union and mostly in the single market after Brexit has been kicked around for several years. Known as the "backstop," the concept is that Northern Ireland could continue trading with Ireland, a member of the EU, without tariffs, border checks or other impediments to the seamless flow of goods between them.

Mrs May, in deference to her coalition partners and fearful that a border in the Irish Sea would be a first step to breaking up the United Kingdom, expanded the backstop to put all of Britain into the bloc's customs union. The EU, in its hunger to strike a deal with Britain, went along with that demand.

But the backstop was anathema to the hardline Brexiteers like Mr Johnson. They said it might delay Britain's exit indefinitely and threatened to leave Europe without any deal if Brussels did not abandon it.

A no-deal Brexit would badly damage the Irish economy. By the British government's own reckoning, a hard border could mean lost jobs, a flourishing black market, roadblocks and civil unrest as people dealt with sudden dislocations. Some predict a revival of the violence in Northern Ireland once known as the Troubles.

"We were meant to have solved all our problems," said Ms Monica McWilliams, an academic and former politician in Belfast who was involved in the Good Friday negotiations. "Then, bang, along comes Brexit, and the whole Northern Ireland issue is back on the table."

Ireland understandably opposes any deal that does not include a backstop, and Europeans are signalling their solidarity. Mr Phil Hogan, the former European agriculture commissioner who was recently appointed Europe's top trade official, said he saw glimpses of flexibility in Mr Johnson's visit to Dublin.

Before his meeting with Mr Varadkar, Mr Johnson spoke of the possibility of an "all-Ireland food zone," in which agricultural products could be traded seamlessly between Northern Ireland and the republic. The European Union would probably balk at such a partial arrangement, but as Mr Hogan put it in an interview with The Irish Times, "the penny is finally dropping."

As Mr Varadkar welcomed Mr Johnson beneath a fluttering Union Jack in Dublin, he gave his guest the same message he got from the opposition and members of his own party during his stormy debut in Parliament the week before: Leaving Europe without a deal is a non-starter.

"There's no such thing as a clean break, or just getting it done," Mr Varadkar said to Mr Johnson, throwing his own words back at him.

For his part, Mr Johnson struck a conciliatory tone, avoiding the fiery language he used in Parliament about the need to get out of Europe, come what may.

"I want to find a deal," he said. "I have looked carefully at no-deal. Yes, we could do it. The UK could certainly get through it. But be in no doubt, that outcome would be a failure of statecraft for which we would all be responsible."