Hamas' deputy foreign minister, Ghazi Hamad, told the radio station Sunday that the group would cease its attacks on southern Israel if Israel halts its military operations against Gaza militants.
Hamad said that "we are interested in calm but want the Israeli military to stop its operations."
The bitter enemies have been locked in heavy fighting in recent days.
Earlier Sunday, Israeli Prime Minister Benjamin Netanyahu threatened a harsher response if militants continue attacking southern Israel with rocket, mortar and missile fire.
THIS IS A BREAKING NEWS UPDATE. Check back soon for further information. AP's earlier story is below.
JERUSALEM -- Palestinian militants in Gaza fired mortar rounds and a rocket at southern Israel early Sunday, the military said, prompting Israel to threaten a harsh response while holding out the inducement of an end to escalating violence.
Israel did not launch immediate reprisals Sunday. Israel's defense minister, Ehud Barak, said if militants in Hamas-ruled Gaza cease their attacks, so would Israel.
Barak, who directed Israel's three-week military offensive in Gaza in early 2009, told Israel Radio he was not eager to launch a large-scale operation in Gaza.
But Prime Minister Benjamin Netanyahu took a more combative tack.
"If the attacks on Israeli citizens and soldiers continue, the response will be far harsher" than it has been, Netanyahu told his Cabinet.
Hamas spokesman Sami Abu Zuhri said militants would cease fire if Israel did. Both sides have accused the other of violating pledges to halt the recent fighting.
No injuries were reported in the attacks Sunday, which followed the most intense single day of fighting between Israel and Gaza militants since Israel's major offensive in the Hamas-ruled territory ended in January 2009. Police said Sunday's rocket hit outside Ashkelon, a city of 120,000 located 11 miles (18 kilometers) north of the Gaza Strip.
But while neither side appears interested in all-out war, the fear is that an isolated incident could easily spark an Israeli offensive because of the combustible situation that has developed over the past month.
The current round of violence escalated last weekend when an Israeli airstrike killed three Hamas militants.
Israel said the men were planning to carry out a cross-border kidnapping. On Thursday, Hamas militants fired a guided anti-tank missile at an Israeli school bus, wounding the two people on board, including a teenage boy who was critically hurt. Several dozen students had exited the bus shortly before the attack.
According to the military, Palestinian militants have launched 120 rockets and mortars into Israel in the past 48 hours. The Israeli retaliation for Thursday's bus attack has killed twelve militants and six civilians, according to Gaza officials.
One of the militants killed was involved in the 2006 kidnapping of an Israeli soldier who has been held in Gaza for the past five years, the military said.
Gaza has been controlled over the past four years by Hamas, an anti-Israel militant group backed by Iran.
Israel invaded Gaza in December 2008 in an attempt to stop years of persistent rocket fire at Israeli civilians near the Palestinian territory. Some 1,400 Gazans, including hundreds of civilians, were killed in the three-week offensive, and 13 Israelis were also killed.
Since that offensive there have been shifts in the balance of power.
On Thursday, Israel made its first use of a new system designed to intercept incoming rockets. The system, Iron Dome, has now successfully shot down eight projectiles aimed at Israeli cities, the military said, though Israeli officials say it cannot provide a complete defense.
Since then, Israel says Hamas has acquired deadlier weapons brought in through smuggling tunnels under the Egypt-Gaza border. Although the militants' arsenal still is dominated by crude rockets made in Gaza weapons shops, Israeli defense officials claim the militant group now has rockets capable of striking Israel's Tel Aviv heartland, 50 miles (80 kilometers) north of Gaza, as well as anti-aircraft missiles and sophisticated anti-tank weapons like the laser-guided missile that hit the bus on Thursday.
Israeli defense officials say Iran is sending weaponry to its Hamas proxy by land and by sea. Last month, Israel intercepted a cargo ship that it said was loaded with weapons, including sophisticated land-to-sea missiles, sent by Iran to Gaza militants.
Hamas and Gaza's other militant groups may have been emboldened by the ouster of longtime Egyptian president, Hosni Mubarak, who shared Israel's interest in crushing Gaza militants on his border. As Egypt prepares for elections, potential leaders have signaled they will take a more lenient approach toward Hamas.