A poll of Conservative Party members has shown almost half do not want a Muslim prime minister.
Some 43% of card-carrying Tories agreed with the statement: “I would prefer to not have the country led by a Muslim”, the survey by YouGov found.
The same study found 45% believe there are “areas in Britain in which non-Muslims are not able to enter” and 67% believe there are areas “that operate under Sharia law”.
Hope Not Hate, the anti-racist campaign group which commissioned the research, described these as “lies” and “Islamophobic myths”.
The poll of 864 party members, carried out this month, also showed just 8% would be “proud of Britain if we were to elect a Muslim as our prime minister”.
It came as Home Secretary Sajid Javid, who is of Muslim heritage, used a live TV debate to encourage all his fellow leadership contenders to hold an inquiry into alleged Islamophobia within the party.
His call follows a series of expulsions and investigations by Tory chiefs into allegations of anti-Muslim comments by party activists and Conservative councillors.
Yet the latest survey found 79% of members do not believe the party has an Islamophobia problem.
And only 15% of members believed the party should be doing more to combat any Islamophobia or other racism within it.
FullFact, the independent fact-checking charity, has found there are no areas where the law cannot be enforced in the UK. The police service inspector has said there are “communities from other cultures who would prefer to police themselves”, but he did not say whether or not any of those are Muslim.
The government has said there “is evidence of a problem” in some cases around the application of Sharia law in the UK but that a better understanding of the issue is needed, FullFact found.
Matthew McGregor, campaigns director at Hope Not Hate, said both Hunt and Johnson should now challenge the views of some party members.
McGregor said both contenders should “state clearly that they accept there is a problem, and confirm their commitment at last week’s TV debate to a full and independent investigation”.
Hope Not Hate has sent both candidates an open letter. The Conservative Party has been approached for comment.