"There is a tendency sometimes among some in the international community to try and understand, to reach agreements, to take a backward step," said the foreign minister.
Russia's special Middle East envoy, Alexander Kalugin, addressed Putin's invitation to Hamas to visit Moscow, saying that Russia would attempt to convince Hamas to change its extremist path, to take up peaceful dialogue, and to recognize Israel's right to exist.
"We want them to respect all of the past agreements to avoid terror attacks," Kalugin told the Russian news agency Interfax.
Livni, who is visiting the United States, is expected to return to Israel Friday afternoon.
In her interview with the New York Sun, the foreign minister warned of a "slippery slope" embarked upon by Russia, which could grant legitimacy and compromise with Hamas.
'Weakness will harm everyone'
"Any weakness… will result in a negative effect - not only for Israel, but also for the Palestinian people and for the international community," Livni said in the interview.
Livni said that Israel would continue to protect its citizen, and will target anyone on their way to commit acts of terrorism.
The foreign minister said there was a difference between an Israeli soldier who prevents a terrorist attack, and a terrorist who targets children in their strollers, youths in clubs, and families eating at restaurants. There is a legal and moral difference here, said Livni.
Meanwhile, U.N. Secretary-General Kofi Annan urged the international community to give Hamas time to change it ways before ruling it out as a partner.
"We are at a very early stage of the game," Annan told reporters in New York.
"Hamas won the election but they have never been in government. They need time to organize themselves," he said.
Annan also told Hamas to listen to the warnings of the international community, to take upon itself the commitments of the Palestinian Authority, to abandon the path of terrorism, and to recognize Israel.
Like Israel, the U.S. government was amazed and furious with Russian President Vladamir Putin's invitation to Hamas. A U.S. official asked how Russia would respond to a U.S. invitation for Chechen rebels to visit Washington.
U.S. anger is further fuelled by the signature of Russia to a Quartet agreement to freeze contacts with a Hamas controlled PA until Hamas ended terrorism, recognized Israel, and abided by agreements signed between Israel and the Palestinians.