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Israeli tank fire killed a Palestinian in the Hamas-ruled Gaza Strip, local medical workers said.
Israel's military had no immediate comment on the cross-border shelling on the northern town of Beit Hanoun on the eve of an Israeli election in which security is a major issue.
Earlier, Israel launched air strikes against what the military described as Hamas outposts in Gaza, saying this was a response to a Palestinian rocket salvo on Sunday.
Palestinians said at least one Israeli missile hit a building used by Hamas police. There were no casualties.
A fragile ceasefire has been in place since Israel ended a 22-day military offensive in Gaza on January 18 that was designed to punish the Islamist Hamas group for cross-border rocket and mortar bomb attacks.
Egypt is trying to secure a lasting ceasefire between Israel and Hamas. Diplomats said the Egyptian proposal includes a prisoner exchange and the initial opening of at least two of the enclave's border crossings.
With his opinion poll lead narrowing ahead of Tuesday's parliamentary election, right-winger Benjamin Netanyahu has pledged to end Hamas rule in the Gaza Strip and proposed focusing peace talks with the Palestinians on economic issues.
His main opponent, the centrist Kadima party leader Tzipi Livni, Israel's foreign minister, said she would continue US-sponsored land-for-peace talks, an outcome favoured by the new Obama administration in Washington.
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