Mrs May, who enraged Conservative supporters by telling them they were seen as the "nasty party", has suggested they are open to accusations of being "racist or bigoted" because they have only one black MP.
The MP, who has been considering entering the leadership race, told an Asian television programme that the Conservative Party could be unelectable because it has failed to reflect the changing face of Britain.
"Take a look at modern Britain, and you will see how much society has changed. A tolerant, liberal, diverse, multi-ethnic, multi-faith society isn't just an aspiration, it is a reality," she said.
"All of us are surrounded by the reality that British society is now made up of different faiths, cultures and races. On television, on the sports field, and in every office and workplace, the face of Britain is no longer white and male. So why when society has changed so much, has the Conservative Party failed to keep pace with those changes?"
Her remarks on The Asian Factor, on the channel ARY TV, will be broadcast on Sunday. Mrs May's sentiments will be applauded by some Tory modernisers who regard her as one of the most outspoken and honest voices in the Tory party. But they are seen by Tory traditionalists as ammunition for the party's opponents.
Last night, Sir Malcolm Rifkind, the shadow Works and Pensions Secretary, echoed the critique, saying in a wide-ranging lecture that the Tories must change their tone on immigration and asylum. He called on them "not to exaggerate the problems" by advocating remedies that might provide good headlines without gaining results.
He said that the party led by him would support any proposals that would be effective in combating illegal immigrants or bogus asylum seekers, but would be mindful of the need to be careful in the language used to debate the issues, given its potential impact on good race relations.
Mrs May said: "Look at our dynamic and diverse society, made up of so many races and religions. And then consider the question, Why doesn't that same British society think we are the party to represent it? It is the ultimate no-brainer! People don't think we resemble or understand them or their priorities."
She added: "It certainly makes it more difficult for us to argue effectively on issues such as controlled immigration, because it is too easy for people to accuse us of being racist or bigoted."
The Tory ballot next Tuesday is expected to reject the proposed change in the voting system for electing a leader, leaving the final choice with the rank-and-file members.
The MPs will have to choose two candidates for the run-off, possibly next January. That would leave the fight over who runs against David Davis, the front-runner. Kenneth Clarke could be edged out by the MPs, which would cause a backlash among the party members.