The vote - by secret ballot - will take place at Westminster on Monday between 6pm and 8pm, with the count to take place immediately afterwards.
It comes after a steady stream of Tory MPs called publicly for the Prime Minister to stand down in the wake of Sue Gray's report into breaches of the Covid regulations in No 10 and Whitehall.
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Meanwhile, the Prime Minister has attempted to deflect attention away from the situation by focusing on the war in Ukraine.
Jacob Rees-Mogg has said that even if Boris Johnson wins by one vote it would be "enough" for him to have a mandate to continue.
A number of other cabinet ministers have also come out in support of Mr Johnson.
Sir Graham said in a statement: “The threshold of 15% of the parliamentary party seeking a vote of confidence in the leader of the Conservative Party has been exceeded.
“In accordance with the rules, a ballot will be held between 1800 and 2000 TODAY MONDAY 6th JUNE — details to be confirmed.
“The votes will be counted immediately afterwards. An announcement will be made at a time to be advised. Arrangements for the announcement will be released later today.”
In order to oust the Prime Minister however the rebels will need 180 MPs, and allies of Mr Johnson made clear he is determined to fight to stay on.
Speaking shortly after Sir Graham made his announcement, Health Secretary Sajid Javid told Sky News: "If there is (a vote) the Prime Minister will stand and fight his corner with a very, very strong case."
Sir Graham said he had informed Mr Johnson on Sunday that the threshold of 15% of the parliamentary party calling for a vote had been passed.
"I have followed the rules that we have in place. I notified the Prime Minister yesterday and we agreed the timetable for the confidence vote to take place," he said.
"He shared my view, which is also in line with the rules that we have in place, that that vote should happen as soon as it could reasonably take place and that would be today."
A No 10 spokeswoman said: "Tonight is a chance to end months of speculation and allow the Government to draw a line and move on, delivering on the people's priorities.
"The PM welcomes the opportunity to make his case to MPs and will remind them that when they're united and focused on the issues that matter to voters there is no more formidable political force."
In his first public statement since the confidence vote was announced, Boris Johnson chose to focus on the war in Ukraine.
The Prime Minister's allies have warned that the Russian invasion is a reason why the leader should not be changed.
Mr Johnson highlighted the UK's supply of rocket artillery systems to Ukraine.
"We cannot stand by while Russian long-range artillery flattens cities and kills innocent civilians," he said.
"The UK will gift the Ukrainian armed forces multiple-launch rocket systems so they can effectively repel the continuing Russian onslaught."
Earlier, former minister Jesse Norman became the latest Tory MP to go public with his call for Mr Johnson to go.
In a scathing letter posted on social media, he said the Prime Minister had presided over "a culture of casual law-breaking" in No 10 and that his claim to be vindicated by Ms Gray's report was "grotesque".
He said the Government's current priorities were "deeply questionable", its policy of offshoring the processing of asylum claims to Rwanda was "ugly", while any breach of the Northern Ireland Protocol would be "politically foolhardy and almost certainly illegal".
"Neither the Conservative party nor this country can afford to squander the next two years adrift and distracted by endless debate about you and your leadership," he said.
"For you to prolong this charade by remaining in office not only insults the electorate, and the tens of thousands of people who support, volunteer, represent and campaign for our party; it makes a decisive change of government at the next election much more likely. That is potentially catastrophic for this country."
Jacob Rees-Mogg said that even if Boris Johnson wins by one vote it would be "enough" for him to have a mandate to continue as Prime Minister.
The Brexit opportunities minister said: "One is enough. That's the rule in a democracy - if you win by one you win."
Pressed on the matter during a Sky News interview, he said: "One is enough, it's no good saying that the rules of the party say something and then behind it unofficially, there is some other rule that nobody knows and is invented for the purpose.
"I obviously want the Prime Minister to get as big a majority as possible, I think that would be helpful and it would close this matter down between now and the next general election, which would be good for the country, good for the Conservative Party, but one is enough."
Asked to be clear on whether this means he believes Mr Johnson would have a "clear mandate" to rule if he won by such a small majority, Mr Rees-Mogg said: "To be absolutely clear, the answer is yes."
He said the Conservative Party wrote the rules for a "straight up and down vote".
But Cabinet ministers have rallied round Mr Johnson - including those who could seek to replace him if he is forced out.
Foreign Secretary Liz Truss said: "The Prime Minister has my 100% backing in today's vote and I strongly encourage colleagues to support him.
"He has delivered on Covid recovery and supporting Ukraine in the face of Russian aggression. He has apologised for mistakes made. We must now focus on economic growth."
Chancellor Rishi Sunak said "the PM has shown the strong leadership our country needs".
"I am backing him today and will continue to back him as we focus on growing the economy, tackling the cost of living and clearing the Covid backlogs," Mr Sunak said.
Deputy Prime Minister Dominic Raab said: "The PM has got the big calls right - securing life-saving vaccines, firing up our economy and standing up to Putin's aggression against Ukraine.
"We need to back him, unite and focus on delivering the people's priorities."