Ms Cherry has emerged as a possible contender for the seat’s SNP nomination after a pointed tweet aimed at Ms Davidson yesterday.
Ex-deputy party leader Angus Robertson and MEP Alyn Smith are also believed to be considering putting their hands up to contest the same seat.
Former SNP MSP Marco Biagi, who represented Edinburgh Central until 2016, said there was “definite warmth” for Ms Cherry thanks to her leadership in the push to revoke Article 50 and stop the Brexit process.
He declared Ms Cherry would topple the Scottish Tories leader from the Edinburgh Central seat if an election contest was held tomorrow.
“From people I have spoken to [in the area] there is a definite warmth for Joanna Cherry, in part because she is better known,” Mr Biagi told The National.
“A lot of people don’t know Angus Robertson grew up in the constituency and went to school in the constituency.
“When people hear that they are more open to him, but at the moment on the basis of familiarity and profile, Joanna seems to be very popular with people I have spoken to.”
Mr Biagi added: “If the election was tomorrow, Joanna would be ahead.
“But the election isn’t tomorrow, so it’s hard to say.”
The next Scottish election is due in 2021, with SNP branches yet to select their candidates.
Ms Cherry has been the MP for Edinburgh South West since May 2015. She is the SNP’s justice and home affairs spokeswoman in the Commons.
Mr Biagi said he would be torn between which of the three SNP candidates to back to take on Ms Davidson.
He said: “Joanna has massive profile right now through her work on the EU, but we won’t know what the situation will be in six months’ time, what sort of bids they will be putting forward or if indeed they all run.
“It’s exciting. It’s aline-up that would flatter a national office.
“I don’t think the last (SNP) depute leadership race had such a range of people with such weight within the party running. Edinburgh Central can be grateful there is such an interest.”