WASHINGTON—Despite months of painstaking effort, senior U.S. officials say they don’t expect Israel and Hamas to agree to a cease-fire and a hostage-release deal before the end of President Biden’s term.

A senior adviser to Israeli Prime Minister Benjamin Netanyahu has presented to the Biden administration a new proposed ceasefire and hostage deal with Hamas, an Israeli official told CNN.

Israel has responded to Hezbollah’s fire with strikes in southern Lebanon, and has struck senior figures from the group in the capital Beirut. The exchanges have killed hundreds in Lebanon and dozens in Israel and forced the evacuation of tens of thousands of residents on each side of the border.

Khaled Meshal, one of Hamas’s most senior officials, said in an interview that the militant group expects to play a decisive role in the enclave when the war is over.