In an email to the United States mission seen by AFP, Kuwait said: "We can not agree to the text put forth by your delegation especially as we are considering a draft resolution that deals with the protection of civilians in the occupied Palestinian territories and the Gaza Strip".
Al Hayya said in a statement early Wednesday that there had been considerable mediation efforts - an apparent reference to Egyptian-led talks - resulting in "a consensus to return to the ceasefire understandings of the Gaza Strip".
In southern Israel, angry residents complained about the renewed rocket fire.
The Israeli military released video which they say shows strikes against military targets in the Gaza Strip.
Israeli officials said Wednesday, though, a cease-fire agreement with Hamas had not been reached.
The restless night, with consistent rocket fire from Gaza on Israeli communities, followed a sharp escalation Tuesday as Palestinian militants in the besieged coastal enclave fired more than 100 rockets and mortars into Israel.
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Several militant projectiles were shot down by Israel's Iron Dome rocket interceptor system, others landed in empty lots and farmland. The boat was carrying patients, students and injured people who were transferred on Tuesday to Israel's port Ashdod, said Israel army. More than 115 people have been killed since protests began on Gaza's border in March. The overnight Hamas rocket fire reached the city of Netivot for the first time since the 2014 war.
Al-Qassam and al-Quds said their "joint response" of rocket attacks from Gaza on the "occupation's military positions ... is a declaration, that these crimes can not be tolerated in any way". The announcement comes amid reports of intense strikes in Israel and Gaza during what has been described as the worst hostilities since 2014.
It said some of the mortars fired were supplied by Iran.
Hamas initially billed the weekly border protests as a call to break through the fence and return to homes that were lost 70 years ago during the war surrounding Israel's establishment.
Among those associated with RSY-Netzer to attend was Nina Morris-Evans, who said: "By Jewishly mourning the lives lost, we were not condoning terrorism, Hamas or violence of any form". Conricus said the tunnel was unusual in that it snaked under Gaza's southern border into Egypt's Sinai Peninsula and then made a U-turn back into Israel.
Highlighting Washington's isolation on the Israeli-Palestinian issue, Kuwait - a non-permanent Security Council member that represents Arab countries - hopes to win a high number of votes in favour of its proposal.
The U.S.is calling on the United Nations Security Council to hold an emergency meeting about the conflict on Wednesday.