Humza Yousaf would not confirm if he will step down if a motion of no confidence in his leadership is successful.
Leader of the Scottish Conservatives Douglas Ross announced at First Minister's Questions on Thursday that he was calling for a vote on Mr Yousaf's tenure.
Asked if Mr Yousaf would resigned should he lose the motion of no confidence, a spokesman said the issue was "hypothetical" and "the business of next week".
Should he stay on if a vote of no confidence is successful he would be in defiance of the will of the Scottish Parliament.
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It followed the end of the Bute House Agreement after days of speculation.
Mr Yousaf met with Patrick Harvie and Lorna Slater this morning at 8am in Bute House to deliver the news.
A cabinet meeting was then held at 8.30am where an insider claimed cabinet members banged on the table in "enthusiastic" support at the First Minister's announcement.
The experienced insider added there had been "no dissent" from the cabinet and that it was the first time they had heard a cabinet table banged in appreciation.
The First Minister is said to have been "reflecting on the decision very, very carefully", despite having pivoted from support of the Bute House Agreement to saying he would scrap it, all within 48 hours.
Mr Yousaf is believed to have arrived at the decision to end the agreement yesterday morning.
The First Minister's spokesman said the BHA had been signed in order to provide "stability" and the vote on the future of the agreement, brought by Greens activists, had undermined that stability.
He said the four-week wait for the vote would continue to put pressure on Mr Yousaf's leadership.
The end of the deal means the SNP has the challenge of the return to being a minority government, as it has been previously in the periods 2007 to 2011 and 2016 to 2021.
Asked if the relationship with the Scottish Greens could be repaired or if there were fears the party might not be placated from the fury shown by Ms Slater on Thursday morning, the First Minister's spokesperson said it was believed that relations would be cordial.
He pointed to Gillian Mackay's buffer zones bill coming before the Scottish parliament next week and said there was an expectation the parties would work together there.
The spokesman said a situation such as this - a motion of no confidence backed by the Scottish Greens - had been "factored in" as a protentional risk by the First Minister.
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