Mr Cleverly told BBC Politics Live on Monday that Mr Farage's Party could split the vote in the Conservative target seats, leading to the election of MPs who could "frustrate" Brexit.
Instead, they will put their "total effort" into contesting seats held by the Labour Party and "Remainer parties".
Not everyone is convinced it will be the game changer on December 12, because Farage still intends to field candidates in seats the Tories hope to gain from Labour and the Liberal Democrats.
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"We see a Tory Party becoming ever more extreme and right-wing, and out of touch with the majority of people in Scotland".
Welcoming the news, Ms Sturgeon said: "The Greens make their own decisions and it's never an easy decision for any party to take, to stand down, and I certainly am grateful that the Greens are thinking carefully about these things".
Brexit Party sources described to CNN that there has been a divide between the party leadership and candidates who didn't want to campaign against Conservatives that were committed to delivering Brexit. Plus, Johnson has more experience in office and is, frankly, more famous and popular than Farage.
The Conservatives are suddenly faced with a much simpler election campaign: the party can shift resources away from seats it had been defending to focus on Labour gains.
Opposition parties were quick to criticise the move, with many saying the Brexit Party was now an ally of the Conservatives.
The Brexit Party - an ephemeral expression of Nigel Farage's overweening ambition to occupy a permanent position in the limelight - is rendered viable as a electoral vehicle only by the failure to give effect to the referendum result.
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Robert Wheal was due to stand as a candidate in the Arundel and South Downs constituency, but has been stood down as part of the move.
The controversial leader also received a lot of praise from Brexit supporters who commended him for putting the "country first".
Four polls released over the weekend paint an even bleaker picture, placing the Brexit party at an average of 7.5%. And, if the assumption that in the absence of a Brexit party, or of UKIP, supporters will tend to vote Conservative holds, then this is both good and bad news for the Tories.
"I couldn't vote for (local Conservative candidate and Cabinet minister) Liz Truss because I do think the Tories have negotiated a awful deal".
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