The news came as Liam Fox, a former Conservative Cabinet minister, accused Mr Johnson of being more like Edward Heath than Margaret Thatcher over tax and spend.
Writing in the ConservativeHome website, Dr Fox said that he did not think that “all is well in the Boris Johnson premiership”.
He added: “For many Conservatives, including myself, the current government smacks too much of ‘big tax, big spend, big state’ – more reminiscent of Edward Heath than Margaret Thatcher.
“The question of the Northern Ireland border remains a thorny subject, and for many a continued border within the United Kingdom is incompatible with the entire ethos of a Conservative and Unionist Party.”
Dr Fox added that it was important to remember “we are 11 years into a period of Conservative government (or at least Conservative led) and it would be more remarkable if we were not having a dip in the polls”.
He added that “governments have recovered from much worse positions than this, as those who remember the more difficult times of the 1980s will attest”.
He said: “Leadership changes can bring about short-term improvements in political fortunes, but the internal wounds can leave longlasting scars, as the political assassination of Margaret Thatcher prove.”
The suggestion was backed by Sir Anthony Seldon, the author of an acclaimed book of British prime ministers from 1721 to the present day, who said that Mr Fox was “completely right” in Mr Johnson’s belief that the state rather than the market is better placed to deal with issues such as levelling up.