LONDON – President Biden is waging a charm offensive with Great Britain’s temporal and royal leaders, battling to preserve the Anglo-American “special relationship” in the face of perceived snubs as the critical question looms of whether to bring Ukraine into NATO.

As if he were reading from a script carefully written in advance, the American president assured the British prime minister, Rishi Sunak, the trans-Atlantic bond was “rock solid” after Mr. Sunak remarked, rather cryptically, “We have a lot to talk about.”

One topic they did not need to cover was the talk in the British press about Mr. Biden’s Irish roots as accounting for why he didn’t attend the coronation of King Charles III in May. Conveniently, the press forgot that President Eisenhower also skipped the coronation 70 years ago of Charles’ mother, Queen Elizabeth II.

Much more to the point, the Brits and the Americans don’t see eye-to-eye on the pleas of the Ukraine president, Volodymyr Zelensky, for admission as a full member to NATO.

Mr. Sunak, receiving Mr. Biden at No. 10 Downing Street, the residence of British prime ministers, presumably spent much of their meeting trying to persuade Mr. Biden to agree to a limited membership for Ukraine after Mr. Biden had made clear he would oppose the whole idea when leaders of the 31 NATO countries meet Tuesday and Wednesday at the Lithuanian capital of Vilnius.

The chances of NATO admitting Ukraine would appear virtually nil considering that all the members have to agree. American diplomats are now scrambling to avoid a showdown in Vilnius after Mr. Biden told CNN he did not believe “there is unanimity in NATO about whether not to bring Ukraine into the NATO family now, at this moment, in the middle of a war.” 

Mr. Biden’s meeting for tea with King Charles at Windsor Castle in the afternoon is likely to be a love-in at which they agree on what they both have to say about climate change. 

No way would anything disturb the tranquility of that exercise in happy talk despite desultory reports in the British press quoting Mr. Biden about his love for the Irish, who were rebelling against British rule when Mr. Biden’s ancestors migrated to America. 

Mr. Biden’s odyssey gets serious again when he presses on to the NATO session at Vilnius.

It will be all the Americans can do to persuade Turkey’s president, Recep Tayyip Erdogan, and Hungary’s prime minister, Viktor Orban, to go along, reluctantly, with membership for Sweden. Mr. Erdogan is demanding Sweden extradite terrorists to Turkey while Mr. Orban is annoyed by Swedish commentaries on the absence of democratic freedoms in Hungary. 

The real question, though, is how to bring about unity on Ukraine short of admitting Ukraine as a member — a step that would be sure to further infuriate President Putin and incentivize Russians to want to prosecute the war ever more fiercely. 

NATO’s secretary-general, Jens Stoltenberg, is engaging in the requisite rhetoric, saying the meeting “will send a clear message” — that “NATO stands united, and Russia’s aggression will not pay.” 

Mr. Stoltenberg is even suggesting that surely Ukraine will become a NATO member — if not now, then in the pretty near future. Mr. Zelensky will press for NATO membership in person at the gathering  at which NATO members should reaffirm Ukraine’s bid while producing what Mr. Stoltenberg called “a multi-year program of assistance to ensure interoperability” and “upgraded geopolitical ties.” 

The Americans are counting on Mr. Stoltenberg to more or less side with Mr. Biden on his calculated reluctance to bring Ukraine into NATO. In return the Americans have endorsed Mr. Stoltenberg for another year as NATO secretary-general, having nixed Britain’s former defense secretary, Ben Wallace. 

Mr. Biden, however, is giving the impression of toughness on Ukraine by approving the export of cluster bombs for Ukraine’s forces after running low on artillery shells.

The White House national security adviser, Jake Sullivan, was confident that Mr. Biden and Mr. Sunak were “on the same page strategically on Ukraine” despite differences on cluster bombs. 

Great Britain is among more than 100 countries that ban the use of the weapons, which can explode into tiny bomblets over a wide area — another step in escalation of the conflict that remains a proxy war in which NATO countries are not committing their own troops.