Speaking to BBC Breakfast on Tuesday, shadow home secretary James Cleverly declined to comment on whether he would put himself forward in the leadership contest, but did not rule out the possibility.
He said he has been focused on “holding the new Labour government to account on my old portfolio, something that’s incredibly important, and that’s what I will be continuing to do, and we will see what happens once those nominations are open.”
Pressed by BBC Breakfast, he agreed that his response was “not a no”.
“The thinking that I and others will have done and will do about the leadership of the party is important and there is a timescale for that, and I’m not going to rush ahead of that timescale,” Mr Cleverly said.
When asked who he would support if he did not put himself forward as a candidate for leader, he said that was “tomorrow’s work”.
The plans allow as many as 11 of the 121 surviving Tory MPs one week to file their nominations for the first round, which closes on 29 July.
Candidates will also be able to use the summer recess, starting next week, to mount their campaigns.
MPs will then begin narrowing down the field through a series of ballots when parliament returns in early September, reducing the field to four by the Tory conference at the end of that month.
The ballot of party members will close on 31st October.
Only members who have been a member for 90 days or more before the ballot closing, and have been active members at the time of the nominations for candidates opening, will be eligible to vote, the party said.
The BBC understands Mr Sunak had wanted to stand down earlier, in order to allow the party to move on.
But it now looks like he could still be in post for Labour’s first Budget – expected in the Autumn.
Mr Sunak said it was in “the national interest” for him to stay in post to allow “a smooth and orderly transition to a new leader of the opposition”.
He said: “This will allow our party to fulfil its role as the official opposition professionally and effectively.
“I believe this is what is best for the Conservative Party and, most importantly, our country.”
Bob Blackman, who as chairman of the backbenchers’ 1922 Committee will oversee the selection of the new leader, said he was “determined that our party will have a respectful and thorough leadership debate”.
He added: “While there are significant debates to be had about our party’s future, we must remember that the country – and our members – want to see us engaged in proper debate, not personal attacks.”