The US representative at the Jerusalem meeting, Assistant Secretary of State David Welch, told reporters that he and his fellow officials had discussed what Blair's role would be, but the meeting itself had ended with no public announcements.
Blair, who steps down as British prime minister on Wednesday, indicated that he was interested in the job, saying in London that he was ready, in principle, to try to help bring an end to the Israeli-Palestinian conflict.
While aides said that Palestinian President Mahmoud Abbas and Israeli Prime Minister Ehud Olmert would support Blair's appointment as Quartet envoy, Hamas spokesman Fawzi Barhmoum criticised the choice.
"Blair has his own black fingerprints in the history of the Palestinian people. He had supported the Zionist occupation's terrorism and massacres against our people," he said, shortly after the meeting.
Some European diplomats have also questioned Blair's ability to garner broad Palestinian and Arab public support because of his leading role in the Iraq war and his close relationship with US President George W Bush.
Many Arabs see the Bush administration as biased against the Palestinians. But Blair, who steps down after 10 years in power, has frequently urged Bush to take a more assertive role in trying to settle the Israeli-Palestinian conflict.
Bush administration officials first floated the idea of appointing Blair to the envoy post in private meetings earlier this year, long before Hamas seized control of the Gaza Strip more than a week ago, western diplomats said.
AP and Reuters contributed to this report