As well as Olmert and Abbas, Blair is due to meet Tzipi Lvivni, the Israeli foreign minister, Ehud Barak, the defence minister and Benjamin Netanyau, the leader of the main opposition.
He is also scheduled to meet Salam Fayyad, the Palestinian prime minister, before he leaves the region.
Al Jazeera's Alan Fisher said he has to set up his office, but the former UK prime minister has said he will "listen, reflect and absorb" in his first meetings.
"Mr Blair is the most qualified person on earth to resolve this issue"
Dave, London, UK
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"What he's going to do next is really what everyone wants to know.
"Whether or not he is going to stay very close to the parameters set by the Quartet," Fisher said. "Or whether he feels he can stretch those parameters and perhaps do more negotiation.
"What the Quartet wants him to do was to help set up institutions in Palestinian areas, and prepare them for government, but those closest to Tony Blair say, if he sees a chance of negotiation, then he won't pass that up.
"He has a very difficult balancing act to achieve."
Jordanian officials said Blair listened to Jordan's perspective on restarting the Middle East peace process. Jordan and Egypt are the only Arab countries that have made peace with Israel.
Dr Azzam Tamimi, director of the Institute for Islamic Political Thought, said that Blair's impact will depend on what kind of mandate he has.
"If his job is to consolidate the authority of Mahmoud Abbas, and further undermine Hamas, he is going to be doomed to failure.
"But if he is willing to build on his experience as a peacemaker in Northern Island, then the scope is very good. In this case, he has to talk to Hamas."
"The decisions required for peace are not going to come from the envoys"
Saed Erekat, Palestinian negotiator
"Tony Blair needs to encourage the Palestinians, to reunite, to come back again, and to work together, otherwise he will not be able to achieve very much."
The visit comes amid diplomatic activity aimed at bringing Israel and the Palestinians back to the negotiating table.
George Bush, the US president, called last week for an international peace conference on the Middle East in autumn.
On Wednesday, Abdul-Ilah al-Khatib, the Jordanian foreign minister and Ahmed Aboul Gheit, his Egyptian counterpart, are due in Israel to formally present an Arab peace initiative that envisions full Arab recognition of Israel in return for lands the Jewish state captured in the 1967 Middle East war.
Israeli and Palestinian officials have welcomed the involvement of Blair, who brings a high-profile and well-respected figure to Middle East diplomacy.
Ahead of Blair's visit, Israeli and Palestinian officials acknowledged Blair's limitations and said a final peace deal could only come through direct talks.
"What I do with the Israelis, what the Israelis do with me, is the main ingredient," said Saeb Erekat, the Palestinian negotiator.
"The decisions required for peace are not going to come from the envoys."
In Gaza, Hamas spokesman Fawzi Barhoum said Blair could not ignore Hamas, which swept Palestinian parliamentary elections last year, otherwise "it will lead to nothing but failure," he said.