David Cameron has condemned xenophobic abuse after the EU referendum amid a growing chorus of concern over intolerance and hostility.

The prime minister told the cabinet the government “will not tolerate intolerance” and it condemned “some of the incidents we have seen across the country over the weekend of intimidating migrants and telling them that they need to go home”.

His comments came as the mayor of London and the UK’s biggest Muslim organisation spoke out against a spike in racist abuse in the aftermath of the EU referendum.

The Polish embassy in the UK said it was shocked at incidents of xenophobic abuse directed at members of its community in the past few days, and the Board of Deputies of British Jews said it was alarmed by reports of harassment and abuse.

Sadiq Khan, who was elected mayor of London last month, said he had put the capital’s police on alert for racially motivated incidents. “It’s really important we stand guard against any rise in hate crimes or abuse by those who might use last week’s referendum as cover to seek to divide us,” he said.

“I’ve asked our police to be extra vigilant for any rise in cases of hate crime, and I’m calling on all Londoners to pull together and rally behind this great city.”

The Metropolitan police promised to investigate any reports of hate crime and abuse.

Khan also warned against the demonisation of people who voted to leave the EU, saying: “It’s also crucial that we don’t demonise the 1.5 million people in London who voted for Brexit. While I and millions of others disagreed with their decision, they took it for a variety of reasons and this shouldn’t be used to accuse them of being xenophobic or racist. We must respect their decision and work together now to get the best deal for London.”

The Muslim Council of Britain urged political and civic leaders to heal divisions exposed by last week’s vote. It said: “In the past few days, we have witnessed shocking manifestations of hate speech not only online but also on our streets.”

The umbrella organisation has compiled a dossier of 100 reported racist and Islamophobic incidents since the result of the referendum was announced on Friday.

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