Israel's military said it had responded by hitting "military targets" including a tunnel stretching into its territory, weapons stores and militant bases.
Islamic Palestinian groups launched about 100 rockets and mortars at southern Israeli communities around the Gaza border.
"Mortars fired by Palestinian militants hit civilian infrastructure, including a kindergarten", U.S. Ambassador Nikki Haley said in a statement on Tuesday. A number of them were intercepted by air defence systems.
Rockets fired by Palestinian militants are seen over the Gaza Strip heading toward Israel, early Wednesday, May 30, 2018.
The spokesman also stopped short of confirming whether Indonesia had barred Israelis from entering Indonesia following the recent violence in Gaza in which dozens of Palestinians were killed by Israeli soldiers.
Israeli officials said Wednesday, though, a cease-fire agreement with Hamas had not been reached.
"After the resistance succeeded in confronting the (Israeli) aggression. there was a lot of mediation in the past hours", al-Hayya said late on Tuesday.
In an early morning interview on Army Radio with Naftali Bennett, he said that no official agreement had been reached and that Israel will continue to act in its own best interests.
In Israeli towns near the frontier, where rocket warning sirens sounded frequently on Tuesday, schools reopened on Wednesday morning.More news: Storm Alberto closes in on Florida, Alabama and Mississippi
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The violent clashes between Palestinian protestors and IDF on the Gaza border with Israel erupted in March when the Palestinians started their rallies called the Great March of Return. Israel and the USA have described Hamas and Islamic Jihad as a terrorist organizations.
Three Islamic Jihad members were killed in an Israeli strike Sunday, and the group vowed revenge.
Islamic Jihad spokesman Daoud Shihab said his faction "does not want an escalation or war". Israel has said its forces have "operated in accordance with the rules of engagement" to protect its border.
Israel's military said it was not seeking an escalation, but warned Hamas, with whom it has fought three wars since 2008.
This follows weeks of Palestinian mass demonstrations near the fence with border. In late March, thousands began weekly protests demanding both a right to return to land that is now inside Israel and a solution to the crisis facing more than 2 million residents.
Israel claims Hamas is responsible for provoking the violence, which the group denies.
The Israeli government did not reply to the Palestinian statements and released no official announcements regarding the ceasefire. Israeli Defense Minister Avigdor Lieberman commented the issue, saying the strikes against Gaza are "only the beginning".
Kuwait first put forward the idea 10 days ago after condemning Israeli forces' killing of dozens of Palestinians during protests on the Gaza border.
"There is a culture of impunity in Israel for crimes against Palestinians", Maliki said.