They had held ”long-standing concerns about bigotry in our politics and our governing party”, a statement from the Muslim Council of Britain (MCB) said.
The warning came as Baroness Warsi, a former Tory co-chair and cabinet minister, said the party “must start healing its relationship with British Muslims”.
In a tweet, she added: “Endorsements from #TommyRobinson & #KatieHopkins & colleagues retweeting both is deeply disturbing. Independent Inquiry into #Islamophobia is a must first step. The battle to root out racism must now intensify”.
Harun Khan, of the MCB, said on Friday morning: “Mr Johnson commands a majority, but there is a palpable sense of fear amongst Muslim communities around the country. We entered the election campaign period with long standing concerns about bigotry in our politics and our governing party. Now we worry that Islamophobia is ‘oven-ready’ for government.
“We understand that the prime minister insists that he is a one-nation Tory. We earnestly hope that is the case and urge him to lead from the centre and engage with all communities.”
The MCB is the UK’s largest Muslim campaign organisation.
Meanwhile, the Labour Against Antisemitism group said the election result – which saw the Conservatives record their best figures since 1987 and Labour their worst since 1935 – was a “damning verdict” on Jeremy Corbyn‘s leadership of what they called an “institutionally racist” party.
The group added in a statement: “The overwhelming reaction of our members to this election result is one of relief. Underlying that emotion, however, runs anger that the British Jewish community has been brought close to serious threat by the complete failure of the Labour movement to deal with the poison of antisemitism.
“The next leader must be prepared to take all necessary measures to deal with Corbyn’s legacy of anti-Jewish hatred, including cooperating fully with the Equalities and Human Rights Commission investigation. Failure to do so will only lead to further electoral catastrophe for the Labour Party.”
Stephen Pollard, the editor of the Jewish Chronicle who been outspoken on the issue of Labour antisemitism, said that the party’s new leader would have to change its culture. “Although it was the leadership that enabled Jew hate, it emanated from members,” he tweeted.
Mr Corbyn was forced to apologise for the antisemitism that has riven his party in an interview with This Morning hosts Phil Schofield and Holly Willoughby, having refused to do so when grilled by Andrew Neil earlier in the campaign. He was “very sorry for everything that has happened”, Mr Corbyn said, but insisted he had dealt with the problem.
In an interview on Friday, Mr Corbyn said: “I inherited a system that didn’t work in the Labour Party on antisemitism. I introduced the rule changes necessary to deal with it and they’re in operation. But I think antisemitism is an absolute evil curse within our society and I would always condemn it, and always do and always will.”
The UK’s chief rabbi, Ephraim Mirvis, said in a Facebook post that MPs now had a duty to tackle entrenched social problems, including within their own parties.
He said: “The election may be over, but concerns about the resurgence of antisemitism very much remain. Islamophobia, racism and other forms of prejudice continue to afflict our communities and, as has been well publicised, even our political parties.
“It is vital that we now bring the country together, ensuring that the voices of people from across our society are heard and respected. We must focus on our shared values and leave all hatred and prejudice far behind us.”
During the election campaign Mr Johnson came under fire for his past description of women wearing niqabs as looking like “letterboxes”, and was accused of personally contributing to rampant racism in Britain. He countered by saying he was “mounting a strong liberal defence of the right of women in this country to wear what they choose”.
On Friday morning as he gave his victory speech, the prime minister made a point of insisting his new government would be a “one-nation” body working “literally for everyone from Woking to Workington, from Kensington, I’m proud to say, to Clwyd South, from Surrey Heath to Sedgefield, from Wimbledon to Wolverhampton”.
The Independent has contacted the Conservatives for comment.