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By Becky Morton
Political reporter
Rishi Sunak has repeatedly declined to say if he agrees with his home secretary's view that multiculturalism has "failed" in the UK.
In a speech on Tuesday, Suella Braverman said multiculturalism was a "misguided dogma" which had allowed people to "live parallel lives".
Asked if she was wrong, the PM told the BBC the UK had "done an incredible job of integrating people into society".
However, he added that "there's always more progress we can make".
Addressing a US think tank in Washington earlier this week, Mrs Braverman said: "Multiculturalism makes no demands of the incomer to integrate.
"It has failed because it allowed people to come to our society and live parallel lives in it... And, in extreme cases, they could pursue lives aimed at undermining the stability and threatening the security of society."
She added: "We are living with the consequence of that failure today. You can see it play out in the streets all over Europe from Malmo to Paris, Brussels to Leicester."
Bishop of Leicester Martyn Snow and local Labour politicians were among those to criticise her comments, with Leicester MPs Jonathan Ashworth and Liz Kendall describing them as "ignorant and offensive".
Last September disorder broke out in Leicester, involving mainly young men from sections of the city's Muslim and Hindu communities.
In an interview with BBC East Midlands Political Editor Tony Roe, the prime minister was asked if he agreed with his home secretary that multiculturalism had failed in cities like Leicester.
In response, he said: "My mum's from Leicester so it's a place that I know incredibly well… I think that this is something that is incredible about this country, is that it is a fantastic multicultural democracy."
Mr Sunak, who is the UK's first British Asian prime minister, added: "We have done an incredible job of integrating people into society and one of the lovely things about getting the job I have as the first person from my background to hold this job, that's a wonderful thing, but it's also not a big deal in our country.
"I think that speaks to the progress we've made over the years and how far we've come and something we should all be collectively incredibly proud of."
Pressed if he believed Mrs Braverman was wrong, the prime minister said: "I'm saying our country has done an incredibly good job of integrating people from lots of different backgrounds.
"It's important that everyone subscribes to British values. That's the thing that unites us and binds us together."
Downing Street has previously said Mr Sunak signed off the home secretary's speech.
Mrs Braverman was also criticised by charities and some Conservatives for comments in her speech that the international asylum system was no longer fit for purpose and that fearing discrimination for being gay or a woman should not be enough to qualify for refugee protection.
The United Nations refugee agency rejected her argument, saying the 1951 Refugee Convention "remains a life-saving instrument" and did not need "more restrictive interpretation".
The prime minister was speaking to local BBC political editors and radio stations in a series of interviews ahead of the Conservative Party conference this weekend.
Earlier, he repeatedly refused to say whether the HS2 high-speed rail line would run to Manchester, amid speculation the second leg of the line is to be scrapped.