The Islamic group Hamas says it is studying a proposal made at negotiations for a halt to fighting in the Gaza Strip.
White House National Security Communications Advisor John Kirby said on Monday that the proposal was presented to Hamas and negotiators are waiting for its response.
Hamas officials expressed gratitude to mediators on Tuesday. They said the group's leadership is responsively studying the proposal.
But they stressed that Israel remains stubborn and has not responded to any of their demands.
Hamas did not mention details of the proposal, but Qatar-based satellite broadcaster Al Jazeera quoted informed sources as saying it includes a return of civilians to the northern Gaza Strip and delivery of 500 trucks' worth of aid supplies into the enclave per day.
But it is unclear whether negotiators can find common ground, as the two sides are sharply divided. Hamas seeks a permanent ceasefire, while Israel wants to eliminate the Islamic group.
On Tuesday, Israeli forces conducted airstrikes and other attacks in wide areas of Gaza. Local media reported casualties, including women and children.
Israel said on Sunday it withdrew troops from the southern city of Khan Younis.
But on Tuesday the military said its airstrikes and other attacks across the southern region the previous day destroyed terrorist sites, and that operations are going on in the enclave's central areas.