"I can say that we are close, and we are on the edge. It may happen, and it may not," he said. Later, Hamdan added, "I believe there is a good chance to have a cease-fire, which can fulfill the needs of both sides."
Regev said Israel is not interested in a "time out," allowing Hamas to regroup after Israeli strikes have done damage. "We want a new reality" in which Israelis don't live under rocket fire from Hamas, he said.
To succeed, negotiations have to be done "discreetly," he said.
Hamdan said Hamas' actions have been "a good lesson for the Israeli government. It's not good to attack the Palestinians, expecting that they will not react against the attack."
Asked whether Hamas would accept Israel's right to exist, Hamdan said the Palestinian people would not consider it without an end to occupation.
The onslaught of rockets fired Tuesday into Israel continued to be met by Israeli strikes on sites in Gaza.
Attacks included one aimed at Jerusalem, one that caused casualties in the southern town of Beer Sheva, and one that injured five Israeli soldiers.
Another rocket hit a civilian building in Rishon LeZion, part of metropolitan Tel Aviv, Israel Defense Forces spokeswoman Avital Leibovich said.
Netanyahu, at a news conference with Ban, said Israel was fighting back with "surgical operations against terrorists at a time when our own population is being bombarded by rocket attacks."
"If we hope to make these tactics illegitimate, they should be condemned in the most forceful terms by all responsible members of the international community," he said. "The moment we draw symmetry between the victims of terror and the unintended casualties that result from legitimate military action against the terrorists, the minute that false symmetry is drawn, the terrorists win."
Speaking later at a news conference with Israeli President Shimon Peres, Ban urged diplomacy and said further escalation would put the region at risk.
"The world is concerned, gravely concerned, at the rising loss of human lives. Further escalation would be dangerous and tragic for Palestinians and Israelis and would put the entire region at risk. I am here to appeal to all to halt fire and restore calm immediately," he said. "Further escalation benefits no one. Now is the time for diplomacy."
Jerusalem Mayor Nir Barkat said Ban was among those who had to take cover when sirens sounded in Jerusalem, warning of an incoming rocket.
Military officials said the rocket landed in an open area of a village.
Other violence Tuesday included 11 Hamas rockets that hit the Israeli city of Beer Sheva, causing casualties. More than 30 rockets were fired into the area, but most were destroyed by Israel's Iron Dome interceptors.
In Tel Aviv, a man with an ax attacked a U.S. Embassy security guard, Israeli police said. The attacker, who also had a knife, was arrested, police spokesman Micky Rosenfeld said.
The IDF again dropped leaflets across parts of Gaza warning residents to evacuate their homes and go to central Gaza City. The leaflets told them which way to go and what part of the city to stay in once they arrived.
The leaflets are part of Israel's efforts to minimize civilian casualties, Israeli officials say.
As a CNN crew was getting video of families fleeing an area in Gaza, an explosion shook the windows of a school, CNN's Arwa Damon reported. The target was a vehicle around the corner, and the strike killed two people, she said.
Another Israeli airstrike hit a building that houses the local offices of Agence France-Presse in Gaza City. No one from the agency was injured, according to AFP.