Ukip's migration spokesman Steven Woolfe and Nigel Farage's former chief adviser Raheem Kassam are the first contenders to throw their hats into the ring for the Ukip leadership contest.
The two men are bidding to replace Diane James, who quit after just 18 days as leader on Tuesday.
The announcements came as Nigel Farage confirmed that he would be carrying on as interim leader until a permanent successor was in place - prompting a furious row within the party.
Mr Kassam had backed Mr Woolfe in the last leadership election, only for Mr Woolfe to be barred from standing after submitting his nomination papers 17 minutes late.
Mr Kassam, who edits the Breitbart website, told the publication he would consider running on a joint ticket with Mr Woolfe in the latest leadership battle.
Mr Kassam said: "I, like ordinary Ukip members, am so tired of the chicanery at the top of this party.
"There is so much corruption. There is so much duplicity. There are so many people shaking hands with one another and then knifing them in private. It has to stop."
"I'm clear about what I stand for: a strong, united Ukip, free of the Tory splitters. I want this party to continue to be the driving force behind Brexit.
"I want us to address the deep cultural and social divides in this country. I want us to champion causes like Justice for Marine A and tackle the scourge of Islamism in our schools, prisons, and communities.
"I want us to become the real opposition and put this feckless Labour Party to bed."
Mr Woolfe also said that would seek to supplant Labour as the main opposition party if he got elected.
But the MEP admitted he had been tempted to defect to the Tories, before deciding to stick with Ukip.
In a statement, Mr Woolfe disclosed he had been "enthused" by Theresa May's start to her premiership but in the end concluded only Ukip could be relied upon to deliver on Brexit.
"Her support of new grammar schools, her words on social mobility and the growing evidence that she is committed to a clean Brexit prompted me, as it did many of my friends and colleagues, to wonder whether our future was within her new Conservative Party," he said.
"However, having watched the Prime Minister's speech on Sunday I came to the conclusion that only a strong Ukip can guarantee Brexit is delivered in full and only our party can stand up for the communities of the Midlands and the North," he said.
"We can replace Labour as the main opposition party. We can build on our remarkable achievements, stand up for the ignored working class and secure a radically different political landscape in Britain for a generation. Under my leadership, that is what we will do."
Even before his declaration, Mr Woolfe, who is an MEP for North West England, had been installed as clear favourite by the bookies to gain the leadership following Ms James's resignation just 18 days after she was elected to the post .
There was speculation that he will be challenged by Suzanne Evans, the former deputy chairman, who has been cleared to stand after her suspension from the party was lifted.
Earlier, the stage was set for what is likely to be a bruising leadership contest when the party's leader in the Welsh Assembly, ex-Tory MP Neil Hamilton, called on Mr Farage to stand aside as interim leader, saying it would be a "mistake" to carry on.
His comments came after it emerged that Ms James had never been formally registered as leader with the Electoral Commission, leaving Mr Farage still in charge.
The commission said it had only received the paperwork notifying it of the party's change of leadership on Monday of this week, and it had not yet been processed as there were outstanding questions regarding the completion of the form.
Mr Farage told the BBC: "I am technically still leader of the party as Diane James's forms did not get processed, so I will continue as the interim leader."
However, Mr Hamilton said the party needed to "grow up and go beyond the Farage era" if it was to move forward, and warned his domineering leadership style would be a "problem" for his successor.
"You've got to have a very thick skin and give as good as you can get; Nigel Farage is the archetype of that - a combination of elephant and rhinoceros - and nobody was going to live up to him and that's another problem for any successor," he told the Press Association.
"But then he has to be the bride at every wedding and the corpse at every funeral so anybody regarded as a competitor for attention is likely to attract his ire."
The party's ruling national executive committee (NEC) will now meet on October 17 to agree a timetable for an election to find a permanent replacement for Ms James.
In her resignation statement released late on Tuesday, she said she was standing down for "personal and professional" reasons, citing a lack of support among the party's MEPs and officials.
It is thought that her husband's ill health may also have played a role, while Mr Farage pointed to an incident at London's Waterloo Station in which she was verbally abused and spat at.
"To be attacked on the station the way that she was was a shock, and with that came a realisation that actually, when you take this job, your life finishes, this is what you are 24/7, there is nothing else," he said.
"I think she looked down the barrel of that and thought 'This is not how I want to live my life'."
He added: "I am sorry for her. It is a very difficult thing for her to have done. As far as the party's concerned, it's not a great day but it is not the end of the world."