Paul Nuttall: UKIP 'mess' will take time to sort out
The infighting was put to Mr Nuttall when he appeared on the BBC's Andrew Marr Show.
The leader said he would have "no problem" with Mr Carswell remaining in the party if he had in fact lobbied in favour of Mr Farage being knighted.
"We have had it in writing now from Douglas Carswell that he lobbied for Nigel Farage to get a knighthood and he spoke to the people involved," he said.
"He has put it in writing, let's see what happens. It'll go to the national executive from here."
He added: "If it's proven that Douglas lobbied for Nigel to get a knighthood...then I don't see a problem."
Mr Carswell told ITV's Peston on Sunday he would "absolutely" fight the next general election as a UKIP candidate.
In his BBC interview Mr Nuttall also said Mr Banks - who wants to be appointed chairman because he says the party is being run like a "jumble sale" - was not "UKIP's major donor".
"A lot of what Arron Banks says I tend to agree with," he said.
"UKIP needs to be professionalised."
He predicted a "big improvement in donations".
"Give me time to sort out this mess," he added.
Mr Nuttall said he had been "hounded" and subjected to "unfair" treatment by the press since he failed to get elected in the Stoke-on-Trent Central by-election, which was held by Labour last month.
During the campaign, he apologised over claims on his website that he had lost close personal friends in the Hillsborough disaster.
He said a press officer had made the mistake but that he took responsibility, adding that he would "stand in a witness stand in a court of law" and confirm he had been at the match when he was 12 years old.
Speaking to Sky News, Mr Nuttall said: "I witnessed things that no child, I suppose, of 12-and-a-half should have to witness.
"But I will say this - I didn't witness the worst of it because thank God my father was an experienced match-goer and he realised pretty quickly that something was badly wrong and I think we were probably one of the first to leave the stadium."
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