The BBC understands Foreign Secretary Liz Truss, Levelling Up Secretary Michael Gove and other cabinet ministers are backing the prime minister as he assesses the scale of the rebellion against his leadership.
Chancellor Rishi Sunak said the public expected government to be conducted “properly, competently and seriously”.
Health Secretary Sajid Javid echoed this, saying the government was not “acting in the national interest”.
The resignations came minutes after the PM apologised for appointing MP Chris Pincher to a government role.
Mr Johnson admitted he had made a “bad mistake” in appointing Mr Pincher to the role of deputy chief whip earlier this year, despite being told about earlier allegations about the MP’s conduct.
His handling of the row has come in for fierce criticism from the opposition and some of his own MPs.