But at the same time, policies aimed firmly at the right-wing vote were being briefed out to Sunday newspapers.
As the Prime Minister pledged to "stick to the course we are on," the Sunday Times reported a crackdown on NHS treatment for migrants (£) and the Mail on Sunday announced plans to withdraw from the European Convention on Human Rights (ECHR).
Sadiq Khan, Labour's shadow justice secretary, said the Tories were "at sixes and sevens" over their direction.
He added: "Not even Margaret Thatcher tried to remove us from the ECHR."
Home Secretary Theresa May and Justice Secretary Chris Grayling are said to have decided to withdraw from the ECHR, which is enshrined in the UK by the Human Rights Act.
In comments that will anger the Liberal Democrats - who are committed to retaining the Human Rights Act, Grayling said: "I cannot conceive of a situation where we could put forward a serious reform without scrapping Labour's Human Rights Act and starting again."
Lib Dem president president Tim Farron told the BBC's Andrew Marr programme the Tories "must be tempted by us" to stay on the centre ground. Describing his party's victory at Eastleigh as a "game changer", Farron added that "it would be a very foolish thing for the Conservative party to lurch to the right".
Cameron has come under renewed pressure from the right of his party after the Tories trailed behind UKIP in Eastleigh.
In an article billed as the Tory fightback, the Prime Minister said he fully understood the concerns of voters impatient for change.
"These people - hard-working, decent, patriotic people - are who the Conservative Party has always been for. We are on the side of those who want to work hard and get on in life.
"But the battle for Britain's future will not be won in lurching to the right, nor by some cynical attempt to calculate the middle distance between your political opponents and then planting yourself somewhere between them."
He added: "It's not about being left-wing or right-wing - it's about being where the British people are. And where the British people rightly are on all these issues is where the Conservative Party is, too."
The move on the European Court of Human Rights - which is a separate institution to the European Union - will be welcomed by Tory MPs frustrated by a series of controversial rulings, including blocking the deportation of the radical cleric Abu Qatada and ruling that prisoners must be given the vote.
Labour accused Mr Cameron of "caving in" to the Tory right over the Human Rights Act.
"It's clear David Cameron's response to his disastrous result in Eastleigh is a big lurch to the right," a spokesman said.
"He is a weak Prime Minister who is caving in to the demands of the right wing MPs in his party."
UKIP leader Nigel Farage said the Tories always promised "jam tomorrow" and claimed their supporters would no longer believe promises from the party.
Speaking on the BBC One Andrew Marr programme, Farage said he did not believe a shift to the right by the Conservative Party would halt Ukip's momentum, despite widespread reports today of Tory shifts on issues such as the Human Rights Act and the European Court.
He said: "The real problem the Conservatives have got is not UKIP. The real problem is their own supporters look at a Conservative Party that used to talk about wealth creation, low tax and enterprise and it now talks about gay marriage and wind farms.
"When these promises are made no one believes them any more."
Tory backbencher Mark Field criticised claims migration from Romania or Bulgaria could somehow be stopped, saying "people are not fools".
He added: "I think even the headlines today.... about getting out of the European Court of Human Rights... we are in government. Either do something now, and call the Liberal Democrats' bluff on this, or stay quiet.
"It's that sort of cynicism, it's just politicians saying words and not doing anything."