The international Quartet for the Middle East comprises the European Union, Russia, the United Nations and the US.
Blair said Obama's commitment to the region, a settled Israeli government and the Arab world demonstrating "a renewed desire for partnership" meant there was "a new sense of momentum towards peace" in the Middle East.
"There is a lot of thinking that will be done by the Israeli government internally, by the Arab world, by ourselves as a Quartet and the American administration, and I think people want to see 'is there a positive way forward we can achieve here'," he said.
Obama, who last month met Binyamin Netanyahu, the new Israeli prime minister, and separately Mahmoud Abbas, the Palestinian leader, will deliver his long-awaited address to the Muslim world in Cairo.
Last month, Hillary Clinton, the US secretary of state, said Washington has stepped up pressure on Israel to halt illegal Jewish settlements on Palestinian land as a step towards reviving the peace process with the Palestinians.
Clinton said Obama "was very clear ... that he wants to see a stop to the settlements".
Blair, who spends "a week to 10 days" each month in the Middle East, was appointed a representative of the Quartet in June 2007 after he stepped down following 10 years as prime minister.
He said there had been progress in the region since his appointment, although the peace process had slowed in recent months.
"I can say that there's been significant progress made but, frankly in the last nine months it's been extremely difficult whilst there has been a situation of virtual political paralysis on the Israeli side, the transition in the US and then of course the divisions on the Palestinian side," he said.