Theresa May has rejected rivals' claims that the next Tory leader must have supported a Leave vote in the EU referendum - saying people want more than "a Brexit prime minister".
The home secretary promised to bring the Remain and Leave sides together and "govern for the whole country".
But Andrea Leadsom and Michael Gove both said the winning candidate must have backed Brexit.
Candidates have been setting out their stalls in a series of interviews.
Mr Gove, the justice secretary, defended his tactics in the face of criticism, telling Andrew Marr it would have been a "betrayal of this country" if he had allowed Boris Johnson to run.
The contest was triggered by David Cameron's decision to step down by October after he was defeated in the EU referendum.
The PM has said he will leave it to his successor to formally announce the UK is leaving the UK under Article 50 of the Lisbon Treaty.
Mrs Leadsom said the UK should "get on with it", but Mrs May told ITV's Peston on Sunday the UK had to finalise its "negotiating stance" first.
The home secretary has a comfortable lead among MP nominations, over her rivals Mrs Leadsom, Michael Gove, Stephen Crabb and Liam Fox.
But she dismissed suggestions others should stand aside to present a unity candidate, saying she wanted a "contest" and that it was important Conservative members "have their opportunity to have a say".
In Sunday newspaper articles, Mr Gove and Mrs Leadsom emphasised the need for the next prime minister to have backed Leave in the referendum.
Mr Gove told The Sunday Times only someone who had backed Brexit would have the "mandate" to lead.
Mrs Leadsom told the Sunday Telegraph that it "would be odd" to have a leader who did not believe in leaving the EU.
In an interview with the Sunday Telegraph, Mrs Leadsom said: "The prime minister resigned because he didn't back leaving, he doesn't believe in it, so it would be odd to just appoint somebody who also didn't believe in it.
"I just don't think that would be right."
She said that if she was to become prime minister, there would be "absolutely zero risk" that Brexit would not happen.
The energy minister also said she wanted to emulate the leadership qualities of Lady Thatcher, who was Conservative prime minister from 1979 to 1990.
She said: "As a person, she was always kind and courteous and as a leader she was steely and determined.
"I think that's an ideal combination - and I do like to think that's where I am."
'My confidence evaporated'
Mr Gove also said the next prime minister should be someone who had campaigned for Brexit.
"It is a matter of democratic accountability," the justice secretary said in The Sunday Times.
"People who voted in a particular way will expect someone who believed in that to argue their case for them...
"If you are going to have a leadership election, the prime minister having chosen to stand down, then the logic is that you need to have someone who backed Brexit and believed in it and argued for it and campaigned for it, as leader in these negotiations".
He also said he would set up a "pentagon" of five senior ministers, made up mostly of those who backed Brexit, to run the EU negotiations.
In The Sunday Telegraph, Mr Gove also explained how his confidence in Boris Johnson - a leading Brexit campaigner who had been regarded as the favourite to replace Mr Cameron, but ultimately did not stand in the leadership contest - had "evaporated".
"That led me to make the difficult decision, at no little cost, to put friendships aside and act in the national interest."
Having "often" declared that he did not want to be prime minister, "I came to the uncomfortable conclusion that I was the person best qualified to deliver that change".
'Consumed by Brexit'
Former defence secretary Mr Fox, who also campaigned for Leave, also spoke to the Sunday Telegraph and said he would increase defence spending if he became leader of the party.
Meanwhile, in an interview with the Mail on Sunday, Mrs May rejected the suggestion the next leader needed to have backed Brexit.
"The government cannot just be consumed by Brexit, there is so much more to do," she said.
And Work and Pensions Secretary Mr Crabb, in a joint interview with his running mate Sajid Javid in the Mail on Sunday, said he had more experience than Mrs Thatcher had when she became leader.
Meanwhile, Scotland's First Minister Nicola Sturgeon has called for "immediate guarantees" on the residency status and rights of European Union nationals living in Scotland.
She has written to Mr Cameron and the five leadership candidates to say EU citizens in Scotland are "understandably anxious and uncertain" about what the referendum result means for them.
Although the UK overall backed leaving the EU, Scotland 62% voted to Remain.
As of Friday evening, Mrs May had secured the backing of 96 of the party's 330 MPs in the leadership context, while Stephen Crabb had 22, Mrs Leadsom had 21, Mr Gove had 18 and Liam Fox had 10.
More than 160 MPs have yet to declare whom they will support.
The candidates will take part in a series of ballots of the MPs, starting on Tuesday.
The two most popular will then go on to a vote of the wider party membership, with the result due on 9 September.