The meeting between the former Scottish Conservative leader and the former UK Prime Minister took place in a hotel in Edinburgh on 8 October.
Mr Blair's 45-minute conversation with Ms Davidson was dominated by the topic of Scotland's constitutional future, it has emerged.
The ex-Labor leader did not meet the Scottish Labour leader Richard Leonard during the trip.
Since quitting as party leader, Ms Davidson has said she is prepared to fight once again for the Union if another independence vote is announced.
Speaking at the Wigtown Book Festival a few days before meeting Mr Blair, she said on the prospect of a second separation poll: "Look, I hope there won’t be a next time.
“I will do what I can to stop that happening, but if it is happening there is absolutely no way that I am going to sit it out.
“This is my country, it’s what I’ve fought for, it’s what I believe in.
“And whether anyone wants me to hold a position or whether they want me to go round, knock doors and hand out leaflets, I’m happy doing both.”
Ms Davidson went on to tell journalist Sarah Smith, who was chairing the event, that she left her leadership job because she was “hopelessly conflicted by Brexit” and also wanted to spend more time being a mother.
A spokesman for Ms Davidson told The Times, "As the MSP for Edinburgh Central, Ruth meets with people from the private, public and charitable sector who have an interest in Scotland's constitutional future."
A spokeswoman for Mr Blair declined to comment.