Livni was speaking after officials in Hamas were quoted as saying that the creation of the new Palestinian unity government and Shalit's release were in no way connected.
Palestinian Authority Chairman Mahmoud Abbas told Prime Minister Ehud Olmert during their meeting Sunday that he would make an effort to secure the release of Shalit as soon as possible, and perhaps before the formation of the unity government.
Livni, currently in Washington where she will meet U.S. Secretary of State Condoleezza Rice, said Abbas now had more influence on Hamas, which is keen to form a coalition with Abbas' Fatah movement.
"Abu Mazen [Abbas] has stated that before establishing a new Palestinian government, he wants Gilad released," Livni told Army Radio.
"If in the past he could say that he was separate from Hamas and had no power to influence them, today his situation has changed. Today, with Hamas wishing to form a government, and, with Abu Mazen moving toward an agreement with Hamas, we expect that the release of Gilad Shalit will be part of this process."
According to Livni, "This is an Israeli demand, and I believe it should be an international demand as well. Now we must see if [Abbas} is capable of keeping this [promise]."
Livni: EU standing firm on Hamas, for now
Speaking to Israel Radio, Livni played down an EU statement last week that said the nascent Palestinian coalition must adopt policies reflecting the long-standing demands of the Quartet
of Mideast peace makers - the U.S., EU, Russia and UN.
The language in the statement has raised concerns in Israel that the EU is easing its commitment to the Quartet's principles as the rival Fatah and Hamas factions try to cobble together a government. Livni said the EU statement did not represent any softening of its commitment to the Quartet conditions, but acknowledged that things could change.
"The Europeans expect any Palestinian government to meet the Quartet conditions. Europe is a partner in the Quartet," Livni said. "At the moment Europe has not changed its position, and I hope it will not do so soon."
Olmert and Abbas met face to face in the Prime Minister's Residence Jerusalem for more than two hours Sunday. According to the Prime Minister's Office, the two leaders agreed to continue meeting on a regular basis in order "to discuss issues related to security, the war on terrorism, and improving the conditions in which Palestinians live."
Senior Abbas aide Mohammed Dahlan said the meeting was difficult and produced no agreements. Dahlan said the Palestinians called on Olmert to ease travel restrictions in the West Bank by removing roadblocks, and to release sick and elderly Palestinian prisoners in Israeli jails, but Olmert made no promises.
The prime minister promised that, beginning at the end of the month, the Karni goods crossing would begin operating on extended hours in order to handle of the transport of goods into and out of the Gaza Strip without delays.
Olmert reiterated his commitment to the two-state solution, in accordance with his Sde Boker address several months ago.
According to PMO sources, Abbas told the prime minister that he supports the three principles laid out by the Quartet of Middle East negotiators, according to which the PA government must recognize Israel, renounce terror and abide by prior agreements.
The PA chairman also accepted Israel's security demands, namely the release of Shalit, a halt to the ongoing Qassam rocket fire, and prevention of arms smuggling into the Gaza Strip.
The prime minister also told Abbas that not only would Israel not recognize the PA unity government unless it agrees to the Quartet demands, Israel would not hold contacts with Fatah ministers serving in the unity government. In essence, Olmert told Abbas not to appoint Fatah officials who are in contact with Israel, in order to prevent the contacts from being terminated.
Olmert also asked Abbas how the $100 million in tax revenues that Israel transferred to the PA in recent months was spent. Israeli officials have said that a large portion of the funds was used to pay the salaries of members of the PA security forces, instead of being used to reform the security forces.
The meeting was Abbas and Olmert's second round of talks in less than a month, and comes weeks after Abbas' Fatah movement and the ruling Hamas reached an agreement at a summit in Mecca to form a unity government.
Hours before the summit, Olmert said Sunday that Israel was willing to treat the 2002 Saudi peace initiative "seriously," and said he supports a regional summit to discuss the plan.