The BBC has refused to allow Boris Johnson to appear on the Andrew Marr Show this weekend unless he agrees to take part in a one-on-one interview with Andrew Neil, amid ongoing tension between the Conservatives and broadcasters.

The prime minister has failed to set a date for his proposed half-hour primetime interview with Neil, despite every other party leader agreeing to do one.

Labour has accused Johnson of dodging one of the toughest interviewers on television, insisting they were operating in the belief that the prime minister was already signed up when they agreed to let Jeremy Corbyn appear on the programme.

“The reason [Johnson] is doing this is because he thinks, like his Bullingdon friends, that they are above the rest of us, that they don’t need to be held to account, they don’t need to be treated like the rest of us,” said John McDonnell, the shadow chancellor.

The Conservatives had said they would allow Johnson to appear on this Sunday’s Andrew Marr Show. A BBC source said this offer had been declined and suggested the broadcaster would not let Johnson pick which shows he wants appear on.

“He won’t be doing Marr until we have confirmed and announced a date for the Neil interview,” said the BBC source.

It is possible the Conservatives could decide to ditch both programmes.

Neil used to be Johnson’s boss when the future prime minister was editor of the Spectator, having been brought in to try to keep the Conservative politician under control following a series of scandals.

In any eventuality it is likely that at least one member of the Johnson family will be on the Andrew Marr Show this Sunday. A source told the Guardian the prime minister’s sister, the broadcaster and pro-remain campaigner Rachel Johnson, is lined up to appear as a guest on the programme.

The row over the prime minister’s public appearances follows his boycott of Channel 4 News’ climate change debate on Thursday in which he was replaced with a melting ice sculpture.

In response the Conservatives formally complained to media regulator Ofcom about alleged bias, threatened to change the public service remit of Channel 4 if they won the election, and sent their own camera crew to follow Michael Gove, the chancellor of the Duchy of Lancaster, as he attempted to take Johnson’s place on the debate.

Changing Channel 4’s remit would require legislation and be difficult to implement, suggesting threats by the Tories may have been to distract from the debate. However, senior figures in the party said they were seriously looking at the channel’s future, at a time when Ofcom is undertaking a review of what public service television should look like in the 2020s.

Johnson took questions from journalists – including those from Channel 4 – at a press conference on Friday morning, in what appeared to be a deliberate bid to rebut claims he is ducking scrutiny. This followed a lengthy LBC interview during which the prime minister refused to say how many children he has.