The talks in Portugal, which currently holds the EU's rotating Presidency, bring together UN Secretary-General Ban Ki-moon, U.S. Secretary of State Condoleezza Rice, EU foreign-policy chief Javier Solana, and Russian Foreign Minister Sergei Lavrov.
The meeting is the first since militant Hamas took control of the Gaza Strip from Fatah last month.
There are not many optimists left who expect any quick solution to the Palestinian-Israeli conflict. The situation is only getting worse, but there is a tentative hope that Blair can restart the peace process.
"Part of the meeting is to define an agenda for Tony Blair, for his new role, says Yahia Said, a researcher at the London School of Economics in Britain. "Clearly, he will bring new energy to the position. It's not an entirely new position -- James Wolfensohn, former head of the World Bank, ran it not extremely successfully."
Said says Blair could also end up failing, but there is hope he will be capable of doing more than his predecessors. "He could be handicapped by Britain's position on the Lebanon war last year," he says. "But in the end of the day, if there is someone who could convince Israel and the United States to undertake some steps to meet the Palestinian halfway, it could probably be Tony Blair."
Tony Blair's Popularity
Today, U.S. Secretary of State Rice said Blair as Middle East envoy could breathe new life into the long-stalled talks between Israel and the Palestinians.
However, Hamas, which controls the Gaza Strip, has already said that Blair is not an honest broker. Said says he expects Blair will try indirectly to find some way to talk to Hamas.
The analyst also believes that Blair is more popular among Palestinians than officials from the current U.S. administration.
"Tony Blair has consistently insisted that there will be no solution to any of the problems of the Middle East without a fair solution to the Palestinian issue," Said says. "He has consistently called for a Palestinian state. So, on one hand he has a consistent record of calling for addressing the Palestinian conflict as a precondition to progress in the Middle East. On the other hand there is a legacy of his positions on Iraq and on Lebanon."
The Quartet itself remains the most active international forum for discussion of the conflict. It was set up as a guardian of the 2002 "road map" for Middle East peace, a three-phase process toward a final settlement of the Israel-Palestinian conflict -- one that was supposed to take place by 2005.