The last shipment of fuel to Gaza by Israel, which is the sole distributor of it to the territory, came before Palestinian fighters attacked an Israeli fuel depot on April 9.
Israel has besieged Gaza since fighters from Hamas, a Palestinian organisation, routed rival Fatah forces there in June.
Fatah, which is led by Mahmoud Abbas, the Palestinian president, is engaged in a peace dialogue with Israel while Hamas refuses to recognise it.
But Ghazi Hamad, a Hamas spokesman, told Al Jazeera on Thursday that there had been "big progress" in negotiations with Egypt towards a temporary ceasefire with Israel.
"We are now waiting for the answer from the Israeli side for the last answer about all ideas proposed by Hamas," he said.
“We are talking about a comprehensive and mutual ceasefire in the whole Palestinian territories.
"I think the key condition for this ceasefire is that Israel should re-open all the crossings, especially the Rafah crossing, in order to allow people and goods to move in and out and to lift the embargo on the Palestinian people. Without opening the crossings, there will be no means for the ceasefire."
UNRWA's latest warning that aid could be suspended comes a day after John Ging, the head of UNRWA in Gaza, said that the aid agency’s fuel supplies were due to dry up.
Robert Serry, UN special co-ordinator for the Middle East peace process, said in a press conference with Ging that it was "wrong for Israel to punish a civilian population for attacks" carried out by Palestinian fighters.
"I call on Israel to restore fuel supplies to Gaza, and to allow the passage of humanitarian assistance and commercial supplies, sufficient to allow the functioning of all basic services and for Palestinians to live their daily lives," he said.
But he called on Hamas "to immediately end attacks against the crossings, whether by it or any other faction or group," to encourage the resumption of fuel deliveries.
"These attacks endanger both international and Israeli civilians, and cannot possibly contribute to Palestinian efforts to ease the blockade of Gaza," he said.
UNRWA delivers aid to about 860,000 of Gaza's 1.4 million population, with the UN World Food Programme delivering food and essential items to an additional 270,000.