Jordan's Ambassador Dina Kawar said she would be meeting with representatives from Arab countries and council members to gauge whether there is support for a "unified text" on advancing Israeli-Palestinian peace.
If a consensus can be reached, a draft resolution could be presented to the council later this month or in January, she said.
"We are going to try to make it before Christmas, if not it will be in January," Kawar told reporters.
"We really want to get everybody on board. That's our intention."
The Palestinians, backed by the Arab League, circulated a draft resolution at the end of September that called for ending the Israeli occupation by November 2016.
The text ran into opposition from the United States and other members of the council, opening the way for the Europeans led by France to begin talks on a separate draft that would set a timeframe for ending negotiations.
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"We will be sitting together and seeing how the possibilities of working with everybody to get as close as possible to a unified text," said Kawar.
Palestinian representative Riyad Mansour told AFP on Monday that he expected the new resolution to be presented by the middle of the month, with a vote to quickly follow.
"The French are moving more and more, trying to bring all the European colleagues together, and I think that eventually they will succeed," he said.
French Foreign Minister Laurent Fabius told parliament last week that a final settlement should be reached within two years.
Fabius is due to meet US Secretary of State John Kerry in Brussels on Wednesday to discuss the French proposal that would also pave the way for an international conference to re-launch the peace process.
It remains unclear whether the United States, which has vetoed UN resolutions seen as critical of Israel, would support a text to re-start the peace process.
Jordan's push at the United Nations came as France's National Assembly voted in favour of recognising Palestine in a non-binding measure similar to those adopted in Britain, Spain and Ireland.
Denmark is also planning to hold a vote while Sweden has taken the additional step of offering recognition of Palestine.Jordan in push at the UN for Palestine resolutionJordan in push at the UN for Palestine resolution