The Quartet of Middle East mediators from the EU, Russia, the UN and US has discussed the appointment of Tony Blair as special envoy for the region.
Mr Blair will hand power to Gordon Brown after 10 years in office
It is expected that Mr Blair will be offered the job after he steps down as UK prime minister on Wednesday.
It was the Quartet's first meeting since Hamas seized control of security from pro-Fatah rivals in Gaza.
Washington has signalled its desire for a role for Mr Blair, but many Arabs see him as too close to the US and Israel.
The Quartet met for three hours at the US consulate in Jerusalem and representatives left without speaking to the media.
Afterwards, in Washington, state department spokesman Tom Casey confirmed there had been talks about the position of envoy, but would not say who might be appointed.
However, Mr Casey noted that Mr Blair would be someone who "would be received well" and be "very successful doing whatever he chooses to do".
Diplomatic sources have told the BBC the delay in the announcement is due primarily to objections from Russia.
Relations between Tony Blair and Russian President Vladimir Putin have been fraught, particularly following the poisoning in Britain of Putin critic Alexander Litvinenko last year.
However, diplomats predict that Russia will come round to Mr Blair's appointment, probably by Wednesday, the day he leaves office, says the BBC's Laura Trevelyan at UN headquarters.
Hamas took full control of the Gaza Strip earlier in June
Mr Blair himself on Tuesday underlined his interest in the job.
"Anybody who cares about greater peace and stability in the world knows that a lasting and enduring resolution of the Israeli-Palestinian issue is essential," he said.
"And I will do whatever I can to help such a resolution come about."
The Middle East situation has become more critical in the last two weeks, as a Palestinian unity deal collapsed between the Islamist Hamas and secular Fatah.
Mr Blair's expected role would be to focus on helping the Palestinians build effective institutions on behalf of the Quartet.
However, he remains a controversial figure in the Middle East, primarily because of the war in Iraq.
He was also much criticised in the region for refusing to condemn Israel's bombardment of Lebanon in 2006.
A spokesman for the Islamist Hamas group said "the experience of our people with Blair was bad" and his appointment "may even make things worse".