Palestinian leaders first made the call for a UN visit in February, but the 15-member council has not answered. Diplomats said the United States leads opposition to such a trip while Israeli-Palestinian talks are frozen. Israel is also opposed.
Eight Security Council members are from the 110 nation Non-Aligned Movement. An envoy from one of them, Pakistan's UN ambassador Abdullah Haroon, said the group would "approach the Security council to respond favourably" to the Palestinian request for a visit.
He told reporters the council must not remain as "bystanders" in the Middle East deadlock.
Haroon said several council nations had responded "favourably" about a possible trip, and that more talks would be held with the United States.
Talks between Israel and the Palestinians have been halted since September 2010. The Palestinian leadership has since stepped up complaints about Israeli settlements in the occupied territories.
Morocco's UN ambassador Mohammed Loulichki said any country which went to the Palestinian territories "will go with a message of hope, a message of peace and a message to encourage all parties to resume as quickly as possible genuine and effective negotiations."
The Palestinian envoy to the UN, Riyad Mansour, said meanwhile that the Palestinian leadership was still considering whether to revive a bid for full membership of the United Nations.
Its application is now stuck in the UN Security Council, where the United States has threatened to veto any vote in favour.
Mansour said the Palestinians could go to the UN General Assembly to get its current observer status upgraded. He said Palestinian President Mahmud Abbas was consulting "with the largest number of Arab countries and friendly countries" on the timing.
Abbas is expected in Paris on Thursday to meet France's President Francois Hollande.
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