Douglas Carswell has called for Nigel Farage to step down as Ukip leader and said the party needs a "fresh face" and an "optimistic" message.
Ukip's only MP told the BBC the party needed to "change gear and to change its management" in order to grow its support beyond its current level.
In response, Farage said Carswell should "put up or shut up".
Carswell, said of Ukip: "We all need to think very carefully as to whether or not we can build beyond the base we have now got without that change. We have gone from being a party with 2%-3% market share to a party with 13%, 14%, 15% market share, and that's a phenomenal achievement, an incredible achievement and no one can ever say that that hasn't been an achievement."
He added the party "needs to change gear and to change its management if it's to go the next level and the Oldham by-election to me said, very clearly, that I think we need a fresh face".
Despite calling for Farage to go, Carswell has said he does not want to be leader himself. He said there are "a lot of talented people in the party" and that "no party is defined by any one person".
Speaking to Sky News, Farage said he was "relieved" Carswell had finally gone public with his call for a change of leadership.
"He’s been having a go at me every single day since the general election, it's just that he’s been doing it in private briefing members officials or the press," Farage said.
"We are a very strongly united party. If you look at NEC of Ukip the MEP group of Ukip, I’ve got terrific support."
He added that Carswell was trying to "sow division where none exists". Ukip's leader said Carswell had "almost no support in the party".
Ukip MEP Patrick O'Flynn, who himself once described Farage as "snarling, thin-skinned" and "aggressive" said the timing of Carswell's criticism could not be worse.
"The biggest event in British politics for more than 40 years - our referendum on leaving the EU - is imminent. For Ukip to instigate a leadership contest would therefore not be sensible and would risk undermining the goal it was set up to achieve," he said.
"Nobody has done more than Nigel Farage to bring about the referendum and in my view he has a major role to play in the campaign and has earned the right to lead Ukip into it. It is time to concentrate on the battle against those who would keep us in the EU, not to be drawn into internal disputes."
Paul Nuttall, Ukip's deputy leader, told BBC Essex dismissed the idea that Farage should go. "Of course he shouldn't and it would be insane for him to do so," he said.