Britain’s foreign secretary, Philip Hammond, told the Telegraph icily about Mr. Johnson: “People who aspire to hold offices of great responsibility do have to show that even under pressure they retain their cool and they don’t step over any red lines.”

Mr. Obama neatly parried the thrust without responding in kind, saying that he thought even Britons would understand that he might find it “appropriate” to have a bust of the Rev. Dr. Martin Luther King Jr. instead.

But the dispute touched deeper chords here about whether Britain can better control its borders and defend itself from terrorism from within or outside the European Union. Mr. Johnson argued that Mr. Obama is “hypocritical,” saying the United States would never cede such control over its sovereign borders to anyone.

Many of those who favor a British exit, or “Brexit,” argue that only outside the European Union can Britain truly control its borders and limit immigration. But behind the anxiety about immigration is not only fear of losing jobs to foreigners, but of potential terrorist acts carried out by Muslim extremists with European Union citizenship and passports who have gone to Syria to join jihad and then have returned to Europe and its freedom of travel.

That freedom is limited by Britain’s refusal to join the Schengen passport-free zone, meaning that all foreigners who wish to enter Britain can be checked. But it is also true that one or more of the men responsible for the November terrorist attacks in Paris, reportedly including the apparent ringleader, Abdelhamid Abaaoud, were able to come to Britain the previous summer and leave again without difficulty.

All of those anxieties have surfaced in what has become a nasty campaign to succeed Mr. Johnson as mayor of London. In both the exit debate and the mayoral race, it is the side that is thought to be losing that has ratcheted up the rhetoric, at times verging on racism and Islamophobia, that has upset many Britons.

The Labour Party candidate for mayor, Sadiq Khan, has been attacked by his Conservative opponent, Zac Goldsmith, for previous appearances alongside Islamic extremists — criticism repeated by Mr. Cameron in Parliament.

The contrast between Mr. Khan, 45, and Mr. Goldsmith, 41, is stark. Mr. Khan, the son of Muslim immigrants from Pakistan, was raised with seven siblings in a public housing unit with three bedrooms. His father drove a London bus, and his mother worked as a seamstress.

Mr. Goldsmith lived in an 18th-century mansion; attended Britain’s most exclusive school, Eton College (where he was thrown out for smoking cannabis); and inherited a fortune from his billionaire father, the publishing tycoon Sir James Goldsmith.

• What is Britain deciding?

  A referendum on June 23 will ask voters whether the country should “remain a member of the European Union” or “leave the European Union.”

• Who is voting?

  British citizens over 18 (although not all of those living abroad are eligible), and residents of Britain who are citizens of Ireland or of the Commonwealth, which includes 53 countries.

• Is this vote final?

  Yes — at least for the foreseeable future. If “leave” wins, there will be a two-year negotiation with the European Union about the terms. It is unlikely to be amicable.

• Has this happened before?

  Britain joined the European Economic Community, the predecessor of the European Union, in 1973. In a 1975 referendum, more than 67 percent voted to remain.

• What is likely to happen?

  Broadly speaking, opinion polls show the population divided evenly. Market analysts and betting shops think that voters will choose to remain.

Were Mr. Khan to succeed in the May 5 election, he would be the first Muslim to lead the city that dominates Britain’s economic, cultural and political life, and where nearly one in eight residents is Muslim.

But his past as a human rights lawyer and activist has been highlighted by Mr. Goldsmith, who describes Mr. Khan as a “threat to London,” accusing him of “appalling judgment.” Mr. Goldsmith has suggested that by speaking on the same platform as those with radical views or who had been accused of supporting terrorists, Mr. Khan gave tacit support to extremism.

In response, Mr. Khan has defended his commitment to human rights. He claims that Mr. Goldsmith wants to divide communities and accuses him of targeting voters, especially Indian voters, along ethnic lines during a “desperate, negative campaign” intended to inflame anxiety about Muslims.

Mr. Goldsmith said that Mr. Khan “has over and over again given platforms and oxygen and even cover and excuses for people who do have extremist views.” He said that he could not trust Mr. Khan “to be in charge of City Hall, in charge of our security, in charge of our police.”

Mr. Cameron recently joined the argument, telling Parliament that Mr. Khan had appeared publicly nine times with Suliman Gani, an imam who Mr. Cameron claimed “supports I.S.,” referring to the Islamic State.

Mr. Khan has acknowledged contacts with Mr. Gani, a constituent. But Mr. Khan said that his support for plans to legalize same-sex marriage prompted Mr. Gani to campaign against him, and that Mr. Khan helped remove Mr. Gani as a local imam.

Mr. Gani called the allegation a “smear on my good name” and described the Islamic State as a “terrorist and inhumane organization.” He also released a photograph of himself alongside Mr. Goldsmith.

Mr. Khan also supported Babar Ahmad, another constituent, in an eight-year battle against extradition to the United States for trial. Mr. Ahmad was eventually extradited and later pleaded guilty to providing material support to terrorism online, but he has been released and is back in Britain.

In an interview, Mr. Khan acknowledged that the accusations may have done him damage. “If I have given the impression, somehow, by being a human rights lawyer that somehow I have similar views to people I abhor, that would be something that I would regret,” he said.

He emphasized that he has a plan to tackle radicalization and has run risks for confronting extremists, including receiving death threats over his support for same-sex marriage.

Part of the background to the Khan debate is not about Islam at all but elements of anti-Semitism in the Labour Party. The party’s leader, Jeremy Corbyn, has been criticized for not doing enough to stamp out anti-Semitic views in the party.

The difficulty is defining the line between anti-Semitism and principled support for Palestine, opposition to Israeli settlements in occupied territory and “anti-Zionism.”

Mr. Corbyn, who has been a fierce critic of Israel, has spoken out against anti-Semitism in the party, but critics say he has not taken tough enough action to expel those who evince it.

The same issue has come up again recently with the election of Malia Bouattia, 28, as the first black Muslim woman to lead the National Union of Students.

For her supporters, her election is a victory for diversity and radical politics. But Jewish students’ groups are alarmed, citing her criticism of the influence of “Zionist-led media,” her description of her Birmingham University as “something of a Zionist outpost” because of its active Jewish organizations and her talk at a meeting that was advertised with a poster featuring Hassan Nasrallah, the leader of Hezbollah.

Within a few hours of her victory, students at the University of Cambridge called for a referendum on whether their union should quit the national body, describing her election as “a horrifying message to Jewish students.” Students at various other colleges, including Oxford, York, Durham, Edinburgh, King’s College London and the London School of Economics, called for their own unions to sever ties with the national union.

Ms. Bouattia has denied any anti-Semitism and has offered to meet with her critics.