The UNSC has not passed a resolution on Israel since 2016, when it condemned Israeli settlement activity under Resolution 2334.
The council has met four times on Israeli-Palestinian violence in the last nine days; twice last week, followed by a public meeting Sunday and a closed door one on Tuesday.
At each meeting, the 15-member body has discussed the issuance of a statement calling for a Gaza ceasefire. The draft bemoans violence against innocent civilians, but does not mention Hamas or its rockets.
The US has blocked the issuance of a statement each time. Along with China, Russia, the United Kingdom and France, the US is one of five UNSC members with veto power.
Such a statement carries less weight than a UNSC resolution, which France is now proposing.
"Overall, the three countries agreed on three simple elements: the shooting must stop, the time has come for a ceasefire, the UN Security Council must take up the subject and we have also called for a vote on a resolution on the subject," the French presidency said in a statement after talks between Emmanuel Macron, Egyptian President Abdel Fattah al-Sisi and Jordan's King Abdullah.
The statement added that the three countries had also agreed to launch a humanitarian initiative for the civilian population of Gaza in conjunction with the United Nations.
France raised the possibility of a UNSC resolution at Tuesday's meeting. Diplomats said US Ambassador to the UN Linda Thomas-Greenfield told the council that Washington's focus would be to continue intensive diplomacy to try and end the violence, noting there had so far been 60 high-level calls by senior US officials, including President Joe Biden.
"President Biden expressed support for a ceasefire," she said.
"We have not been silent and neither have you," Thomas-Greenfield said, according to a UN diplomat familiar with her remarks.
"With regard to further Security Council action, we must assess if any given action or statement will advance prospects for ending the violence," she said. "We do not judge that a public pronouncement right now will help de-escalate."
French diplomats said they believe that a UNSC resolution could raise pressure on the parties to end hostilities, adding that any action in New York would complement other diplomatic initiatives and that they think there is a way to make the United States "evolve."
Separately, the UN General Assembly plans to hold a debate on the matter Thursday, at the requests of the Palestinian Authority, the Arab Group and the Organization of Islamic Cooperation.