US Secretary of State Condoleezza Rice joined the cavalcade of Arab and Western diplomats converging on UN Headquarters in New York Tuesday for an emergency Security Council session on Gaza. Arab delegations planned to introduce a draft resolution at a ministerial-level Security Council meeting, scheduled for Tuesday evening, that would call for an end to the Israeli "bombardment" of civilians - a demand underscored by strikes Tuesday on two UN schools that killed at least 30 people, including children, according to UN officials. Amr Moussa, the secretary-general of the Arab League, told reporters after a closed meeting of Arab ministers Monday that "the sticking point for us is what is happening to civilians." "Now it is our turn to insist on balance in the resolution," he added, referring to the failure of the Security Council to agree on a statement calling for an immediate cease-fire after the US vetoed it as "unbalanced" against Israel. US State Department spokesman Sean McCormack said the United States could only support an immediate cease-fire resolution that was not time-limited and that addressed the three core elements of a proposed US cease-fire: an end to Hamas rocket fire, a crackdown on smuggling into Gaza, and a return to the 2005 agreement to open crossings into the Strip. "We would like to see the violence end today," McCormack said. "But we also want to see it end in a way that is sustainable and durable." Israeli envoy Gabriela Shalev, who told The Jerusalem Post earlier this week that Israel would not accept a Security Council resolution that did not hold Hamas sufficiently accountable for its continuous shelling of Negev towns, was expected to be present at the Security Council meetings, according to her spokeswoman Mirit Cohen. Rice also planned to hold bilateral meetings in New York with Palestinian Authority President Mahmoud Abbas, as well as Arab and European foreign ministers, to curry support for a US cease-fire initiative proposed Monday. In addition to Abbas, Rice was expected to see Turkish Foreign Minister Ali Babacan, whose country favors a cease-fire ensured by international monitors, as well as her British and French counterparts, David Miliband and Bernard Kouchner. Meetings with Arab officials were expected on Wednesday. UN Secretary-General Ban Ki Moon joined in the shuttle diplomacy, traveling to Washington for a previously scheduled "farewell" lunch with outgoing US President George W. Bush and his wife Laura. Ban was expected to discuss "the importance of bringing the violence to a permanent halt," according to a spokeswoman who briefed reporters. Moussa and the foreign ministers of Jordan, Lebanon and Morocco planned meetings with envoys from other Arab nations, as well as Ban and Security Council members. The talks at the UN coincided with a frenzied round of diplomatic meetings in the Middle East Tuesday, including visits by French President Nicolas Sarkozy and Quartet envoy Tony Blair. AP contributed to this report.