Jordan, last month, circulated the draft text, which fixes November 2016 as the deadline for Israel to withdraw from the territories it has occupied since 1967.
It also calls for the recognition of a Palestinian state with borders based on the 1967 division of territory.
The Palestinian Authority announced on Sunday its intention to present, through Jordan's UN representation, the resolution at the Security Council and to put the resolution to a vote as early as Wednesday.
Palestine's UN mission, however, said it was more likely that the text would be submitted on Wednesday, meaning that it is not certain when it will be brought to vote.
The motion requires a “yes” vote of at least nine of the 15 council members to pass, but it can be vetoed by any of the permanent members, including the U.S., a staunch ally of Israel.
On Monday, Israeli Prime Minister Benjamin Netanyahu met U.S. Secretary of State John Kerry in Rome in a bid to ensure American support in blocking any UN move setting a deadline for Israel.
Neither official commented on the discussions that took place at the Villa Taverna, the residence of the U.S. ambassador to Italy.
However, American sources said that Kerry had stopped short of accepting Israel’s demand for a U.S. veto of the resolution, telling Netanyahu that Washington would continue to attempt mediation in the Middle East conflict.
"It's premature to talk about what we would do in a hypothetical situation at the UN. We’re talking to the parties; we’re hearing what their views are," an official from the U.S. State Department told The Anadolu Agency.
"We're engaging with the other stakeholders and trying to stay coordinated and to work to achieve the goals that we all share, preventing an escalation of the situation on the ground," the source said.
Jordan's UN envoy Dina Kawar said she had not received any requests yet regarding action on the Palestinian draft.
Meanwhile, an alternative UN Security Council proposal which calls for a resumption of peace talks between Israelis and Palestinians within two years, is also being discussed.
Kerry is said to be determined to seek support for the French-led proposal to present a compromise resolution at the Security Council. This resolution would restart negotiations, and promote a two-state solution without setting any deadline for Israeli withdrawal from the occupied Palestinian territories.
The U.S.-brokered direct Palestinian-Israeli talks came to a halt in April because Israel refused to release a group of Palestinian prisoners despite earlier pledges to do so.