The nominations for leadership hopefuls - expected to include Mr Smith and Ms Eagle - will open on Monday July 18, according to a leaked timetable.
Labour-commissioned analysis said Mr Corbyn needed nominations from MPs to stand again.
"There will be plenty more to come to his successor, don't worry about that", he declared.
"He is reminding me of the Black Knight in Monty Python's Holy Grail", he said.
When asked why he is also challenging former Shadow Business Secretary Angela Eagle for the leadership, he said: "I think we need the next generation of Labour men and women to step up and put our stamp on this party".
McCluskey said it would be undemocratic to force Corbyn to secure the backing of Labour MPs before he could be allowed to stand, and accused the parliamentary party of launching a "squalid coup".
Incumbent Jeremy Corbyn is entitled to be on the ballot in the Labour Party leadership election, the National Executive Committee of Britain's main opposition formation decided on Tuesday.
There was speculation that the Labour Party could formally challenge the vote through the courts last night, though leading moderates played that down.
"So I'm on the ballot paper. and we will be campaigning on all the things that matter".
Mr Corbyn said: "I have been elected, last year, 10 months ago today, with a very large mandate".
"Jeremy Corbyn is unable to provide the leadership this huge task needs".
In Bolton, North East MP David Crausby and several senior Labour councillors have joined voices calling on Mr Corbyn to resign - although union bosses and many local members disagree.
But Mr Corbyn also said he hoped there would not be a legal challenge to the NEC decision.
Earlier this year, Foster attacked the beleaguered party leader for downplaying alleged anti-Semitism within the party.
Mr Smith described Mr Corbyn's refusal to step down in the face of an overwhelming no confidence vote as "selfish".
"I refused to have any part in discussions, which have been destructive, from a small group of people on the right who, just like those on the left, it seems to me, are now prepared to let Labour split".
"I wouldn't say this is the death knell of the Labour Party - it's not - but it means we are entering very hard dark days ahead", he said.