The former ministers both resigned from Theresa May's government in protest at her position on Brexit.
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.
If Mr Gove formally enters the race, it would bring the total confirmed number of candidates to eight.
International Development Secretary Rory Stewart, who has also confirmed his intention to run for the leadership, hit out at current front-runner Boris Johnson, who suggested during a conference in Switzerland that he would be prepared to see a no-deal departure in the autumn.
Speaking to Sky News, Mr Stewart accused the former foreign secretary of dishonesty and said he would not be able to serve in a government led by Mr Johnson.
"I sat down with Boris Johnson two weeks ago and he said to me he would not be pushing for a policy of no-deal Brexit. I left the room feeling that we had an understanding."
"I now understand, from what he said yesterday, he is going to try to crash us out of the European Union at the end of October... that is not being straight with people."
Health Secretary Matt Hancock, who confirmed on Saturday that he was running, also suggested he was opposed to a no-deal departure.
Mr Hancock said leaving without a deal was "not an active policy choice that is available to the next prime minister" because MPs have already voted against it.
Following Mrs May's announcement that she would step down as Conservative leader on 7 June, the party has set out the likely timetable of the contest to replace her.
Nominations for candidates close in the week beginning 10 June.
Tory MPs will then begin a series of knock-out votes with the view to reducing the field down to the final two candidates.
From 18 June, the party hopes the two contenders will tour the country attending hustings, facing questions prior to the final ballot of members in mid-July.
If that timetable is followed, it is likely Mrs May's successor as Conservative leader and prime minister will be in place by the time parliament rises for its summer recess at the end of July.
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