Edinburgh South-West SNP MP Joanna Cherry and SNP MEP Alyn Smith have also been floated as possible bidders to fight Edinburgh Central for the party at the next Scottish Parliament elections - not due until 2021.
All three have been approached by local activists about standing for the seat where the SNP finished just 610 votes behind Ms Davidson at the last general election in 2017.
But Mr Robertson was the only one who confirmed he was considering going for the nomination.
He said: “I’m hugely grateful for the encouragement by SNP members in Edinburgh Central to stand and I’m thinking seriously about it.”
Mr Robertson, who grew up in Stockbridge and went to Broughton High School, lost his Moray seat to the Tories in 2017. He recently set up pro-independence research organisation Progress Scotland and he and his wife Jennifer are expecting their first child in June.
He said: ”At present I’m very focused on the imminent arrival of my first child and the continuing work of Progress Scotland.”
But he added: “I think the SNP has a great chance of winning Edinburgh Central, but only with a very hard-fought campaign, which will require the commitment of a candidate based and working here over the next years.”
Ms Cherry, who is a QC and the SNP’s justice and home affairs spokesman at Westminster, responded to reports linking her to Edinburgh Central by tweeting: “Happy to take on Ruth Davidson in any forum. True that my Edinburgh South West seat to be abolished under Boundary review and many people approaching me for Edinburgh Central. Tory Party in total disarray on Brexit and riven with division so who knows what’ll happen before next general election.”
The Gorgie/Dalry part of her Westminster seat is in Edinburgh Central at Holyrood.
A spokesman for Ms Cherry said she had been targeted by the Tories at the 2017 election as part of their strategy to remove leading SNP figures like Alex Salmond and Angus Robertson, but she won with a reasonable majority.
He said: “She is very much committed to Edinburgh South-West and is expecting there will be an election at some point and she will fight it and she’s fairly confident she would win it.
“She’s happy to continue being a strong voice in Westminster holding the government to account.
“But members of Edinburgh Central branch are conscious that in the future her seat is due to be abolished and are actively seeking her out to consider standing.”
Mr Smith was linked to the seat as long ago as December. But he said: “I’m 100 per cent committed to stopping Brexit which is the best interests of Edinburgh Central, Edinburgh and Scotland and don’t have time for other people’s speculation.
“I’ve made clear if Brexit does come to pass I want to be where I’m useful to the party and the cause. Beyond that I have no plans.”
Alison Dickie, education vice-convener on the city council, fought the seat for the SNP in 2017 but was not saying if she wanted to stand again.
She said: “The Scottish Parliament selection process has not even opened yet.
“My own focus is right now on absolutely doing my best job for the folk of Southside/Newington and for education across the city.
“I’m hugely conscious there is a wealth of talent among SNP members in this city and I want to do everything I can to encourage anyone, everyone, to consider standing. When the process does open I will make my own decision then too.
“I know amazing people who are not elected at the moment and I want to see many more of them coming forward.”
Labour finished third in Edinburgh Central in 2017 and the Greens were fourth.