Conservative leader Kemi Badenoch has accused Nigel Farage of “fakery” over Reform UK membership numbers, after his party claimed they had surpassed the Tories in signed-up members.

Mrs Badenoch said Reform’s counter was “coded to tick up automatically”, but Mr Farage said he would “gladly invite” a firm to “audit our membership numbers” as long as the Conservatives do the same.

The row comes after a digital counter on the Reform website showed a membership tally before lunchtime on Boxing Day ticking past the 131,680 figure declared by the Conservative Party during its leadership election earlier this year.

When the figure was announced, Mr Farage said it was an “historic moment”.

However in a thread on X later on Thursday, Mrs Badenoch said it was “a fake” and used a clock emoji to say that it was “coded to tick up automatically”.

She added that “we’ve been watching the back end” of the counter “for days”

Mrs Badenoch added: “Farage doesn’t understand the digital age. This kind of fakery gets found out pretty quickly, although not before many are fooled.”

There were 131,680 Conservative members eligible to vote during the party’s leadership election to replace Rishi Sunak in the autumn, but Mrs Badenoch claimed in her thread that “the Conservative Party has gained thousands of new members since the leadership election”.

In response to the thread, Mr Farage said that the “Conservative brand is dying” under Mrs Badenoch’s leadership, and added: “We will gladly invite one of the Big 4 firms in to audit our membership numbers as long as you do the same.”

The official Reform X account also posted an image that it said included a “screenshot of our internal membership numbers”, which appeared to show figures at more than 134,000.

In a post on X on Thursday evening, Mr Farage said that “over 5,000 people have joined Reform today”.

A research briefing published by the House of Commons Library in 2022 said comparing party membership numbers can be “difficult”, saying there is not a uniformly recognised definition of membership, or an established method to monitor it.

Luke Tryl, director of the More in Common think tank, similarly told the PA news agency it is an “opaque” process.

Speaking before Mrs Badenoch made her accusations, Mr Tryl said: “Parties are notoriously opaque about this sort of thing”.

He described party membership as “very opaque and murky as a metric anyway”.

On Reform, Mr Tryl said one of the challenges for the party will be whether membership converts to campaigners.

He told PA: “There is no doubt Reform had a very good autumn. I think they capitalised off some of Labour’s early mistakes, but also the fact the Conservative brand is still struggling. They’ve clearly got momentum.”

Discussing Reform’s membership, he said: “We know that lots of Reform’s most vocal supporters are very online.

“Do those people who are very online and joined up, do they also go out and pound the streets, deliver leaflets, canvass, that sort of thing?

“That remains an open question.”

Reform was set up as a limited company and in September Mr Farage announced that he would change the ownership structure so that it would be owned by members.

“I no longer need to control this party,” he said at the time.

In a video posted on X, he said: “We will change the structure of the party from one limited by shares to a company limited by guarantee, and that means it’s the members of Reform that will own this party.”