Matt Hancock is understood to be considering quitting the race to be the next Tory leader – and prime minister – as other contenders work out how best to take on frontrunner Boris Johnson.
BBC political editor Laura Kuenssberg said the health secretary could make a decision within hours, after getting 20 votes in the first ballot of Tory MPs.
That put him in sixth place in the race, well behind Mr Johnson on 114.
The two contenders left after more MPs’ ballots will go to a party-wide vote.
Mr Johnson received more than double the number of votes of his nearest rival – Foreign Secretary Jeremy Hunt – in Thursday’s first ballot, putting him on course to secure one of those final spots.
Three contenders – Mark Harper, Andrea Leadsom and Esther McVey – were knocked out and it is not clear to which candidates their supporters will transfer their votes.
Mr Johnson, the former foreign secretary and London mayor, said he was “delighted” to win the first ballot but warned his campaign still had “a long way to go”.
Further ballots are scheduled to take place next Tuesday, Wednesday and Thursday to whittle down the contenders until only two are left. The process could be speeded up if anyone drops out.
The final pair will be put to a vote among the 160,000 members of the Conservative Party from 22 June, with the winner expected to be announced about four weeks later.
All of Mr Johnson’s remaining rivals have signed a joint letter committing to taking part in televised leadership debates on Channel 4 and the BBC on Sunday and Tuesday, in an effort to put pressure on Mr Johnson, who has not agreed to do so.
Speaking to BBC Radio 4’s Today programme, Mr Hunt said the frontrunner should be “a little bit braver” and participate in the debates.
“What would Churchill say if someone who wanted to be prime minister was hiding away from the media?” he asked.
International Development Secretary Rory Stewart, who received 19 votes in the first leadership ballot, told the Today programme: “It’s really important for the legitimacy of the prime minister that this [leadership race] doesn’t take place in a smoke-filled room but that the public is able to see the candidates.”
Sources close to Mr Johnson say they are still in discussions with broadcasters.
One of his allies, MP Priti Patel, told Today there would be “plenty of opportunities for questioning”.
“The focus now is on the parliamentary process,” she added.
Mr Johnson has previously been criticised by some of his rivals for not taking part in media interviews during the campaign.
The leadership race has so far been dominated by Brexit and arguments over whether a deal can be renegotiated with the EU by 31 October, and if talking up a no-deal Brexit is a plausible promise.
The UK’s next prime minister
On Tuesday 18 June, BBC One will host a live election debate between the Conservative MPs still in the race.
If you would like to ask the candidates a question live on air, use the form below. It should be open to all of them, not a specific politician.