All five Conservative leadership candidates have said they will support an independent inquiry into allegations of Islamophobia within their party.
Sajid Javid challenged the other candidates on Tuesday's BBC TV debate to commit to an external investigation and the others appeared to agree to it.
Jeremy Hunt said racism was "not restricted to any one political party".
Ex-party chairwoman Baroness Warsi said it was "important" the promise was kept by whoever becomes prime minister.
She has been calling for an independent inquiry into "institutional" Islamophobia in the Conservative Party.
On the BBC TV debate, the candidates were pressed by an imam to accept that "words have consequences" amid claims that the Conservatives have failed to tackle Islamophobia in the party.
Referring to Donald Trump's string of attacks on London's Muslim mayor Sadiq Khan, the home secretary said politicians should be "brave enough" to call out Islamophobia wherever it came from.
Foreign Secretary Mr Hunt told BBC Radio 4's Today programme: "I think we should have an independent inquiry because the cancer of racism and prejudice is not restricted to any one political party.
"We have been very vociferous calling out Jeremy Corbyn and anti-Semitism and if we are going to do that, and I think we are right to do that, then we have to be whiter than white ourselves."
Environment Secretary Michael Gove told the BBC that there were people who need to be "rooted out" of the Conservative Party over Islamophobia.
"We need to be absolutely resolute in tackling racism and prejudice of all kinds," he said.
"Absolutely there are people in the Conservative Party who we need to make sure appreciate the consequences of their actions - there are people who need to be rooted out of the party."
In response to the question from Abdullah Patel on the TV debate, leadership frontrunner Boris Johnson said he was "sorry for the offence" his comments about veiled Muslim women looking like "letter boxes" and "bank robbers" had caused, and mentioned his great-grandfather was a Muslim.
Baroness Warsi, who was the UK's first female Muslim cabinet minister, said: "It's really important that whoever becomes PM keeps this promise.
"It was made on national TV, so I hope they will.
"Secondly, [I hope] that they genuinely appoint someone who is independent and who is trusted really."
Meanwhile, concerns have been raised about allegedly anti-Semitic social media posts from Mr Patel's now-deleted Twitter account.
BBC Radio 5 Live presenter Nicky Campbell, who interviewed Mr Patel on his breakfast show, apologised on Twitter, saying the imam's social media comments were "extremely disturbing" and they "should have checked".
The BBC defended its vetting process, saying in a statement that "one individual reactivated a public twitter account he had previously deactivated" following the debate, resulting in the tweets not being visible during the background research process.
"Had we been aware of the views he expressed there he would not have been selected," the statement said.