“We want a clear and forceful intervention by the Quartet so the two sides implement their obligations under the road map,” al-Maliki said during a speech at a Madrid political forum.
Al-Maliki said Abbas would also push the US to ask Israel to ease checkpoint restrictions in the West Bank.
The European Union, Russia and United Nations form the Quartet with the US.
Negotiations between Israel and the Palestinians were re-launched at a conference in the US city of Annapolis last November and the US president aims to conclude a comprehensive agreement by the end this year, just before he leaves office.
Abbas confirmed he would meet Bush for further talks at the Egyptian Red Sea resort of Sharm el-Sheikh on May 17.
“We would also gladly welcome Israeli Prime Minister Ehud Olmert at the summit,” he added, “but we do not know whether this will be the case”.
Abbas wants the US to press Israel for a framework agreement for a final settlement to the conflict.
However, negotiations have been stalled by differences over continued Israeli settlement building, violence in Gaza, and rocket-firing by Palestinian fighters against Israel.
Meanwhile on Wednesday a senior Syrian politician told Al Jazeera that Olmert has reportedly offered to withdraw from Syria’s Golan Heights in exchange for a peace deal with Damascus.
Buthaina Shaaban, Syria’s minister for expatriate affairs, told Al Jazeera that the offer was put forward through Recep Tayyip Erdogan, Turkey’s prime minister, although Israeli officials have not commented on the report.
On Monday, Hamas, the rival Palestinian faction to Abbas’s Fatah party, said it was ready to accept a Palestinian state within pre-1967 Middle East War borders, but not formally recognise Israel.
“We accept a state on the June 4 line with Jerusalem as capital, real sovereignty and full right of return for refugees, but without recognising Israel,” Khaled Meshaal, the exiled Hamas political leader, said.
Meshaal made the remarks following talks with Jimmy Carter, the former US president, in Syria.
The comments were dismissed by the US and by Israel, whose prime minister refused to meet Carter during his visit to the region.
A US state department spokesman said “nothing has changed” in the group’s stance.