Tories at Holyrood were left reeling in August when Ms Davidson - who was being touted within the party as a future first minister - abruptly resigned her position in the wake of Boris Johnson entering Downing Street.
Jackson Carlaw, her deputy, took on the top job in Scotland but has angered some colleagues after this week backing Mr Johnson’s position on Brexit.
There is growing concern among Conservatives that a widely expected snap election at Westminster could see the party lose many of the constituencies it won north of the Border in 2017.
Party members will gather in Fife next week to discuss their next move and hear how the process of electing a new leader could be sped up, The Times reports.
Among those considering leadership challenges are Tory MSPs Jamie Greene, Brian Whittle, Michelle Ballantyne, Liam Kerr and Miles Briggs.
Mr Carlaw left some of his colleagues furious on Sunday after he claimed it would be “far more damaging” to continue with the “endless drift”, and said he would back the Prime Minister in leaving the EU on 31 October, even if there was no deal agreed with Brussels.
It comes just two months after Ms Davidson, who resigned as Scottish Tory leader in August, said her party “won’t support” a no-deal Brexit.
“My preference is for a negotiated way out. It always has been. But I’m not someone who has argued that the roof will fall in on the world in an alternative scenario if we have properly prepared for it,” Mr Carlaw said.
A source told The Times: “Obviously Jackson is keen to stay on for as long as possible. On the other side interim leadership doesn’t provide the stablility we require. There are too many things going on.”