Mr Davis said polls showing Mr Cameron was still in the lead were "broadly" true but he claimed "they're changing fast" as the gloves came off in their contest for the leadership.
Mr Cameron claimed that the Tories would lose the next election, if they chose Mr Davis as their leader on 6 December. "There is a big choice opening up here - do we go down a sort of core vote, right-wing agenda playing the same tunes that we played," he said on the BBC AM programme. "Frankly, if we play the same tunes, we will end up with the same position in the charts - second."
Mr Davis will hit back today with a speech on social justice aimed at demonstrating that he can reach poorer voters in spite of his promises on tax cuts.
A senior member of the Davis camp said: "This game is not over. We are keeping the pressure up on Cameron and setting the agenda. There is nothing old-fashioned or right wing about tax cuts because they will help people at the bottom."
Mr Davis will attack the law on adoption which forbids grandparents from adopting their grandchildren. He will say: "With more mothers out to work there is more childcare delivered by grandparents than ever before. But the Government has failed to catch up with this change. There are cases where a child has lost contact with one parent and the other may sadly be unable to manage but that child is put up for adoption even when the grandparents would be willing to take on the responsibility.
"Excluding grandparents from taking on family responsibilities is an appalling ageist prejudice which damages some of the most vulnerable children in our country. We must change this."
He will also call for faster court punishments for young offenders, and for charities to take the lead in supporting drug addicts.
"Commentators talk about the wristband generation," he will say. "I want to win the wristband generation for the Conservative Party. This is the generation who wear the 'make poverty history' wristbands. They display their intolerance of racism with their white and black bands. The blue bands have raised money to highlight awareness of bullying. This generation wants a government that helps their neighbour."Reuse content