Labour has accused Ukip of trying to cover up a longstanding commitment to privatising the NHS, with Ed Miliband saying Nigel Farage cannot escape his comments about wanting companies to run the health service.

The attack represents an escalation in tensions between the two parties in the face of Ukip’s growing threat to Labour in the north of England. Until recently, both Labour and the Conservatives had held off from direct attacks on Farage and his party, preferring to dismiss their influence. But Labour has taken a more confrontational stance in recent weeks, with Miliband saying it was time to “level with Ukip”.

Farage has been forced to clarify Ukip’s health policy in recent days after a video emerged of the Ukip leader saying in September 2012 that Britain should move to a privately run insurance-based system. Farage said the party now believed healthcare should be free at the point of use and dismissed Labour’s “mud-slinging”.

But Miliband said the tape, uncovered by the Guardian, showed “what Nigel Farage really thinks about the NHS – he wants to privatise it”. He added: “Farage can twist and turn but he cannot deny it.”

Andy Burnham, Labour’s health spokesman, has now written to the Ukip leader asking why anyone should believe that his denials are “anything but an opportunistic attempt to cover up your and your party’s longstanding conviction that the NHS should be privatised”.

He asked Farage directly whether he told the meeting in September 2012: “I think we are going to have to move to an insurance-based system of healthcare. Frankly, I would feel more comfortable that my money would return value if I was able to do that through the marketplace of an insurance company than just us trustingly giving £100bn a year to central government and expecting them to organise the healthcare service from cradle to grave for us.”

Burnham pointed out that Paul Nuttall, the Ukip deputy leader, had congratulated the government on “bringing a whiff of privatisation into the beleaguered National Health Service” and that the new Ukip MP Douglas Carswell in 2012 called for an open market in healthcare contracts.

The shadow health secretary then challenged Farage to instruct Carswell to support a Labour backbench bill to stop the privatisation of the NHS, which is due to come to the House of Commons on 21 November, a day after the Rochester & Strood byelection.

Asked by Channel 4 on Thursday whether he still backed an insurance-based system, Farage said: “No, we dropped that, well, we haven’t dropped it, we haven’t accepted it.”