Livni, who heads the ruling Kadima party, said that Israel should work with moderate Palestinians and Arab states for peace, in contrast to her election rival Benjamin Netanyahu, the leader of Likud, who proposes talking to the Palestinians only for economic reasons.
"The dove is on the window sill," Livni said.
"We can either slam the door or let it in. The choice is in your hands."
Livni's speech came on a day when at least five people were injured after an Israeli missile was fired at a car in the town of Rafah in the southern Gaza Strip.
"We targeted a car that carried rocket launchers, they fired rockets and we targeted them," an Israeli army spokesperson told Al Jazeera.
The Israeli military also confirmed it had bombed half a dozen locations in the Gaza Strip overnight.
Palestinian witnesses reported huge explosions. The targets included an abandoned police station in northern Gaza, and suspected smuggling tunnels in the south near the Egyptian border.
The military said the attacks were the beginning of a new wave of raids over Gaza, but did not elaborate, Al Jazeera's Hoda Abdel Hamid reported from Gaza City.
The Israeli military and rescue services said on Sunday that Palestinian mortars hit the village of Nahal Oz, next to the Gaza border fence, wounding two Israeli soldiers and a civilian.
Al-Aqsa Martyrs Brigades, the military wing of the Fatah faction led by Mahmoud Abbas, the Palestinian president, told Al Jazeera that it carried out the attacks.
The Hamas faction seized control of the Gaza Strip from Fatah forces in June 2007, but several other Palestinian groups, including the Brigades, continue to operate in Gaza.
Israel, however, holds Hamas responsible for all rocket fire coming from Gaza.
Israel and a Hamas-led group of Palestinian factions declared separate ceasefires last month after a 22-day Israeli offensive on the Gaza Strip that killed at least 1,300 Palestinians, many of them women and children.
Thirteen Israelis were also killed, three of them civilians, by Palestinian rockets and mortars.
Earlier, Ron Kampeas, an Israeli political analyst, said the underlying motive of Israel's recent raids on Gaza was that Kadima wanted "to show it is as hawkish as [their rival party] Likud".
He told Al Jazeera that Netanyahu appeared likely to win the February 10 election and that Livni was now trying to keep her job as foreign minister in a coalition government.
Kampeas added that it remained to be seen whether Netanyahu, if he won, would form a coalition with Avigdor Lieberman, who is seen as further right than Netanyahu, which "could mean an even more hardline stance towards Hamas".