Dominic Raab and Andrea Leadsom have become the latest Tory MPs to formally announce their intention to succeed Theresa May.
Environment Secretary Michael Gove is also reported to be in the race for the Tory leadership, despite not yet formally declaring his candidacy.
Dominic Raab, the former Brexit secretary, has written in the Mail on Sunday saying he will put his hat in the ring to offer an "optimistic Conservative vision" and "fight for a fairer deal on Brexit".
Andrea Leadsom, whose resignation as Leader of the House of Commons on Wednesday played a role in precipitating Mrs May's resignation announcement, has told the Sunday Times she has the "experience and confidence" to "lead this country into a brighter future".
While both say they will attempt to renegotiate the existing withdrawal agreement with the EU, both also commit to ensuring the UK leaves the EU with or without a deal by 31 October.
"We should keep the arm of friendship extended to our European neighbours," Mr Raab writes in the Mail on Sunday.
"Over the long-term, both sides will want to build a new partnership. But we must also calmly demonstrate unflinching resolve to leave when the extension to negotiations ends in October - at the latest."
Mrs Leadsom told the Sunday Times she would put "significant effort" into trying to find a "deal that we can all live with" but added: "I think any policy needs to be that that is the date of our departure."
According to The Sunday Telegraph and The Sunday Times, Mr Gove has told MPs he would be the best candidate to take on Labour leader Jeremy Corbyn.
He has reportedly talked himself up as a "unity" candidate with the "vision" and "grip" over government departments to deliver Brexit.
Supporters of the environment secretary expect him to publicly announce his candidacy on Sunday night, The Sunday Telegraph reports.