JERUSALEM — Israel and Hamas went back and forth on Sunday over proposals for a new cease-fire in the fighting in the Gaza Strip, and Israel sought to bolster its claim that its forces were not responsible for the deaths of 16 Palestinians reportedly killed in an attack on a United Nations school.

Palestinians who brought their dead and wounded relatives to a Gaza hospital after the attack on Thursday said that hundreds of people who sought shelter in the school had gathered in its courtyard, believing that buses were on the way to take them somewhere safer. Then a number of munitions fired by Israeli forces hit the school, they said, killing and wounding scores of people.