Conservative senator Cory Bernardi is not ruling out quitting the Liberal Party and depriving the Coalition of a key vote in the Senate as a protest against Malcolm Turnbull's ascension to the prime ministership.

Senator Bernardi could form his own right-wing party modelled on the British anti-immigration party UKIP, defect to Family First or even sit as an independent.

Conservative News Corp columnist Andrew Bolt has raised the possibility of Senator Bernardi defecting. Bolt is also a strident critic of Mr Turnbull, largely because the newly installed Prime Minister is a moderate and progressive Liberal.