The military says Palestinians have launched over 110 rockets and mortar shells at Israel since the offensive ended Jan. 18.
"If the rocket fire from Gaza continues, it will be answered with a painful, harsh, strong and uncompromising response from the security forces," Olmert said, speaking at the start of the weekly Cabinet meeting in Jerusalem.
Many Israelis believe the Gaza operation ended too soon, leaving Hamas in power in Gaza and seemingly undaunted. One of them is hard-line Likud leader Benjamin Netanyahu, Israel's incoming prime minister. He is working to cobble together a new ruling coalition and is likely to replace Olmert within weeks.
Israel and Hamas have failed so far to reach a long-term truce through Egyptian mediation, and sporadic violence has persisted. So far, Israel has responded to the near-daily rocket fire by striking smuggling tunnels under the Gaza-Egypt border through which Hamas has smuggled some of its weaponry.
The heightened possibility of renewed hostilities in Gaza came ahead of a week of intense - and possibly crucial - diplomatic activity.
Monday will see top diplomats from around the world convene in Egypt for an international conference on the strip's reconstruction, with the Palestinians seeking $2.8 billion in aid to repair the devastation left by the Israeli offensive.
A string of high-profile foreign visitors has visited Gaza since the offensive, in a sign of increasing international involvement in the long-isolated territory. Former British prime minister Tony Blair, currently an international Mideast envoy, was touring Gaza on Sunday - his first visit since being appointed to the job in 2007.
Last week European Union foreign policy chief Javier Solana made his first visit to the territory since Hamas' 2007 takeover there, and U.S. Senator John Kerry toured Gaza the week before. None met with officials from Hamas, ostracized by most of the international community as a terrorist group.
International aid will not be able to move ahead without a firm truce between Israel and Hamas but the Egyptian-mediated talks between the sides have yielded no results so far.
Hamas wants Israel to open Gaza's blockaded border crossings, without which any major rebuilding will be impossible. Israel says it will do so only after Hamas releases an Israeli soldier it has held since 2006.
Another stumbling block is the feud between Hamas and the more moderate West Bank-based government of President Mahmoud Abbas.
Most of the international community shuns Hamas as a terrorist organization and will not funnel aid through the group, preferring instead to deal with Abbas.
The billions of dollars in donor funds hanging in the balance have served to spur new talks between the two Palestinian factions in an attempt to reach a power-sharing agreement. Those talks are currently under way in Cairo.
Hamas and Abbas' Fatah movement have been at odds for years, a spat that culminated in Hamas' bloody rout of Fatah in Gaza and its takeover of the territory.
The new U.S. secretary of state, Hillary Clinton, is making her first Mideast visit to attend the Gaza donors' conference, at which the U.S. is expected to pledge $900 million. She will then continue to Israel and the West Bank.
Clinton will arrive in Israel at a time of political uncertainty. Olmert is currently a caretaker prime minister in the aftermath of last month's national election, which saw gains by Netanyahu's Likud and other hard-line parties.
Netanyahu's new government could consist of an alliance of hard-line and Orthodox parties opposed to the establishment of a Palestinian state, which could set Israel up for a clash with Clinton and the U.S. administration.
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