Zapatero's office said Blair had insisted there was sufficient political will to bring about peace and stability in the Middle East which he was convinced would come to a two-state settlement.
Blair takes up his role as special envoy amid a renewed US push for peace in the Middle East.
Condoleezza Rice, US secretary of state, Ban Ki-Moon, the UN secretary general, Javier Solana, the EU's foreign policy chief, and Sergei Lavrov, Russia's foreign minister, whose countries and organistations together make up the Quartet, will all attend the Lisbon talks.
Hamas rejected
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"Mr Blair is the most qualified person on earth to resolve this issue"
Dave, London, UK
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At a press conference shortly before the Quartet meeting was due to be held, the US and EU again announced their refusal to deal with Hamas.
Both Rice and Luis Amado, the Portuguese foreign minister whose country currently holds the rotating EU presidency, rejected dealing with Hamas, even as critics questioned whether the stance could compromise Blair's work with the Palestinians.
Rice said: "Hamas, I think, knows what is expected for international respectability."
Speaking for the EU, Amado said: "I see no conditions at the moment to engage [in] new relations with Hamas without a new position from them."
The meeting will also be the first high-level Quartet meeting since Hamas took control of the Gaza Strip, forcing out the Fatah faction led by Mahmoud Abbas, the Palestinian president, which is now concentrated in the West Bank.
The split in the Palestinian leadership has placed another obstacle in the way of a peace deal, but it has also prompted Israel and the West to seek ways to support Abbas, whose West Bank-based government is the only recognised by the Quartet.
While Blair is enthusiastic about his role in the peace negotiations, critics say his limited mandate to help the Palestinians develop institutions and the rule of law, along with his orders not to deal with Hamas, will make it difficult for him to achieve a breakthrough.