But today Blair categorically stated that the meetings are "not about my returning to the front line of politics".
"I have made it abundantly clear that this is not possible," he said in a long, rambling statement, in which he also announced the launch of a new "not-for-profit institute" (nb. "not a think tank").
Since he left office in 2007, Blair has focused on work in the Middle East and Africa.
Today he said the as-yet-unnamed institute will "build a new policy agenda for the centre ground together with the networks which link people up, and allow a reasonable and evidence based discussion of the future which avoids the plague of social media-led exchanges of abuse", he said.
"This platform will have a policy unit which will draw on the best ideas and practical solutions, building partnerships with other organisations, in the public policy and private spheres, so that those in the frontline of politics have a bigger and better policy agenda to reflect upon; and a networking capability to join like-minded people up."
Brexit doesn't necessarily mean Brexit
Last week Blair came under fire (and in doing so spectacularly announced his return to British politics) after he said Brexit could still be halted if concerns around its impact continued to increase.
In an interview with the New Statesman, he said Brexit can be stopped "if the British people decide that, having seen what it means, the pain-gain cost-benefit analysis doesn’t stack up".