Mr Sunak met with the former Labour prime minister as he spent around 12 hours in the United Arab Emirates for the climate change conference.
The Telegraph understands that while their discussion did not last for more than a couple of minutes, it touched on the politics of the Middle East as turbulence in the region continues amid the Israel-Hamas war.
Mr Sunak and Sir Tony briefly discussed the situation on a day that saw a week-long pause in hostilities come to an end as fighting resumed between Israeli troops and Hamas terrorists.
Britain joining the United States in the invasion of Iraq has come to define Sir Tony’s 10-year premiership, with a backlash to the decision continuing to overshadow many other aspects of his decade in power.
Sir Tony has continued to take a keen interest in foreign policy since leaving office, not least through his think tank the Tony Blair Institute.
The Institute has controversially continued to advise Saudi Arabia in the wake of the murder of the journalist Jamal Khashoggi, an act it has strongly condemned.
It came as Mr Sunak also discussed the Israel-Hamas war with Qatari and Jordanian leaders on the sidelines of the main Cop28 programme.
Speaking to broadcasters on Friday, Mr Sunak said: “We’ve been consistent that we want to see sustained humanitarian pauses so that more aid can get into the people of Gaza but also the hostages can come out.
“Those are critical ingredients. And as we’ve said, everyone needs to adhere to the terms of these agreements.”
A Downing Street spokesman said Mr Sunak had thanked Amir Sheikh Tamin bin Halad Al Thani, the Qatari ruler, for “facilitating” a humanitarian pause during their meeting.
The spokesman added: “The Prime Minister reiterated that Hamas had demonstrated that it could not be a partner for peace and could have no future in Gaza.”