The Conservative leadership hopefuls have been invited to a BBC televised debate to test their views on Brexit, putting pressure on the leading candidates to take part.

The broadcaster said it wanted to host the event in mid-June but it is understood the candidates were not consulted about the format, which would involve a clash between all those left in the race at that point – likely to include Boris Johnson, Dominic Raab and Michael Gove.

One supporter of Johnson said the frontrunner was not invited to take part before the announcement was made, although he had been approached by Sky News and ITV about whether he would be prepared to do a head-to-head debate if he was one of the final two candidates.

The frontrunners in the contest have the most to lose by doing television debates in the early stages of the race when they are trying to get through the rounds of votes by MPs, so it is possible some may refuse to take part.

The BBC also said it wanted to hold a second event, in a Question Time format hosted by Fiona Bruce, at the point where the contest moves on to a vote by Conservative members on which of the two finalists to make prime minister.

This would give the leading two candidates the chance to submit themselves to audience questions. The BBC also wanted the two remaining candidates to be subject to interviews by the political journalist Andrew Neil in separate standalone programmes.

After the BBC announcement, Sky News released its own plans for a live head-to-head debate as part of the Conservative leadership contest. It said it would host the two final candidates in the battle for No 10 in front of a studio audience made up of Conservative voters, moderated by presenter Kay Burley.

The leadership contest will formally start on 10 June but all except the top four to six candidates are expected to drop out after the first couple of rounds of voting by Conservative MPs.

Calls for Tory Islamophobia inquiry as equality body launches Labour antisemitism investigation – live news Read more

Before this Friday’s deadline for applications, an 11th candidate, James Cleverly, announced he had joined the race for the party leadership.

Facebook Twitter Pinterest James Cleverly in Westminster on Tuesday. Photograph: Leon Neal/Getty Images

Cleverly, a junior Brexit minister and former deputy party chairman,said in a letter to constituents: “We cannot bring the country back together unless the party of government is united, and the party cannot unite if it is led from its fringes.

“I believe the case for Brexit is still valid, and I have not wavered in that belief. But I have never been blind to the complexities of the process and I have always been uncomfortable with those who offer artificially simple solutions.

“It would be best for the UK to leave the EU with some form of deal, and the EU must now recognise the need for flexibility as the current deal has been rejected by our parliament.”

Kit Malthouse, a former deputy mayor to Johnson, was also a surprise addition to the race, having decided he was the man to deliver the so-called “Malthouse compromise” on Brexit, which was named after him. The compromise – replacing the Irish backstop with alternative arrangements, or negotiating a longer-term transition to a no-deal exit – has been backed by Conservative leavers and remainers but was rejected as unworkable by Downing Street.

He raised eyebrows on Tuesday by suggesting the government could buy up lamb chops to feed to schoolchildren in the event of no deal.

Malthouse told LBC: “Something like 80,000 tonnes of sheep meat is exported every year. Now if we go out with no deal, they think that will be significantly affected, so what do we say to them? Do we say, look, what can we do to make sure that the lamb makes it on to British dining room tables, that probably equates to four to five lamb chops per person per year – could we use it in hospitals and in schools or could we compensate farmers for the disruption this would cause?”

Facebook Twitter Pinterest Kit Malthouse after announcing his bid. Photograph: Isabel Infantes/PA

The pair join Johnson, Raab, Jeremy Hunt, Gove, Andrea Leadsom, Esther McVey, Sajid Javid, Matt Hancock and Rory Stewart in a crowded field of candidates.

Others who may yet declare include the hardline Eurosceptic Steve Baker, Stephen Barclay, the Brexit secretary, Penny Mordaunt, the defence secretary, the Treasury minister Jesse Norman, the former development secretary Priti Patel and Graham Brady, until recently the chair of the 1922 Committee.

It comes as No 10 confirmed that May had promised the EU that the UK would not seek to reopen the withdrawal agreement, as a condition of its extension of article 50, until the end of October. The promise, written into the article 50 extension, undermines the pledges of Conservative leadership candidates to secure a renegotiation in the coming months.

Asked about the legal status of the promise, a No 10 spokesman said: “The legal text that was approved at EU council with the prime minister was that we will not be seeking to reopen the withdrawal agreement. That was part of the process under which the extension was approved until 31 October.”

Rory Stewart hits out at Tory rivals over 'suicide bomber' remark Read more

But Hunt, the foreign secretary, who is the leading candidate among MPs, stressed his confidence in being able to achieve a renegotiation during interviews on Tuesday. He said: “The only solution to the extremely difficult situation we are in, and I don’t want to pretend there is an easy way through this, is to change the withdrawal agreement.”

He added: “We need to have a new negotiating team, with someone from the Democratic Unionist party, the [Tory hard-Brexit] European Research Group, someone from Scotland and Wales so that the union side of these issues is properly thought through.”

He also wrote a Telegraph article saying that a no-deal Brexit would be “political suicide”, as he did not believe parliament would allow the UK to leave without a deal on 31 October, and it would force an early election.

Johnson, the frontrunner, has said he would seek to renegotiate May’s deal and leave at the end of October whether or not he has managed to achieve that. But he has not been specific about whether this would involve opening the withdrawal agreement.

Raab, who launched a campaign video emphasising his commitment to “fairness”, has said he would like to agree a “legally binding exchange of letters to give the UK a clear exit from the so-called backstop”. However, it is not clear whether this would involve reopening the text of the withdrawal agreement.