The new Conservative chair, James Cleverly, has been strongly criticised after apparently reneging on a promise given for an independent inquiry focusing on Islamophobia in the party.

Boris Johnson backed the proposal along with the four other candidates still in the race when it was raised by Sajid Javid during the BBC Tory leadership debate in June.

In an interview on Sunday, Cleverly said there would be an independent inquiry but he did not explicitly confirm that it would be be specific to Islamophobia rather than a general inquiry into racism.

Subsequently, a party source told the Guardian that the party was considering options for “an investigation into all types of prejudice and discrimination”, rather than the Islamophobia-specific one that the Muslim Council of Britain has been calling for.

A spokesman for the MCB said the party needed to clarify exactly what was being proposed, and it would be a travesty if Johnson did not honour the promise given during the BBC debate.

“There are over 100 incidents of Islamophobia that we have identified yourselves in the past year alone,” the spokesman said.

“The Conservative party’s approach to Islamophobia has thus far been one of denial and obfuscation. This needs to change.”

The Labour party chairman, Ian Lavery, said: “Boris Johnson promised an inquiry into Islamophobia in the Tory party but he’s backtracking. You can’t trust a word he says.”

During the BBC debate, Javid, who at the time was home secretary and is now chancellor, managed to bounce his rivals into agreeing to an inquiry by challenging them all to say whether or not they accepted the idea. All of the others, including Johnson, appeared to endorse Javid’s proposal.

In a subsequent interview during the campaign, Johnson seemed to resile from this commitment, claiming that what he had agreed to was “a general investigation into all types of prejudice and discrimination, including antisemitism”. That now appears to be the CCHQ position.

In his interview, Cleverly said that before any inquiry went ahead the party would need to have a definition of Islamophobia.

This year the government was criticised for refusing to accept a definition of Islamophobia drawn up by an all-party group of MPs that said it was “a type of racism that targets expressions of Muslimness or perceived Muslimness”. The government said it wanted to consider the matter further because that definition was not widely accepted.

The MCB has been calling for an inquiry into Islamophobia in the Conservative party for more than a year, claiming that multiple incidents show that party officials are not doing enough to reprimand or remove those who express Islamophobic views.

A YouGov poll this year suggesting that 54% of Conservative party members think Islam is “generally a threat to the British way of life” has also been cited as evidence that Islamophobia is entrenched in the party.

Asked about this finding, Cleverly told Sky that the Tories did not share their membership data with YouGov and that therefore there was some doubt as to whether those surveyed for the poll were all party members.

He said the party took a “robust line” with people who said inappropriate things.