Abbas, who initiated the telephone conversation, thanked Medvedev for the humanitarian aid Russia has delivered so far, and the Russian president vowed that the assistance would continue.
Medvedev ordered on Monday Foreign Minister Sergei Lavrov and Emergency Situations Minister Sergei Shojgu to urgently work on getting more humanitarian aid to Palestinians, the press service said.
Russia sent some 60 tons of humanitarian aid, including foodstuffs, medicine, bedding and tents, to the Gaza Strip on Saturday.
The Israeli military began on Monday a new stage of its ground operation in Gaza, working to clear the Palestinian region of insurgents and terrorist infrastructure, Israeli media reported.
So far, 534 Palestinians have been killed and at least 2,500 injured in the Israeli assault, which became a ground offensive on Saturday after a week of airstrikes. One Israeli soldier has been killed and four civilians died as a result of Palestinian rockets fired in response to the Israeli bombing.
Alexander Saltanov, deputy foreign minister and presidential envoy, met with Israeli Foreign Minister Tzipi Livni on Sunday to try and prevent a humanitarian catastrophe in the region which is home to some 1.5 million Palestinians.
"The central topic under discussion was the sharp worsening of the situation in Gaza and its surroundings. There was a comprehensive exchange of views on possible approaches to overcoming the current crisis. Special focus was paid to the grave humanitarian crisis in the region," Russia's Foreign Ministry said in a statement.
Livni told the Russian envoy, "Israel is conducting a campaign against terror which strikes at its citizens. Other states in the world do likewise, and sometimes also enjoy the direct support of other states. Israel is not asking any other country to fight in its stead - this is our responsibility towards our citizens - but I expect the support and understanding of the international community for the actions that Israel must take."
During a telephone conversation with Israeli Prime Minister Ehud Olmert on Sunday Medvedev expressed Russia's "serious concern" over the numbers of civilian deaths on both sides and urged "to stop bloodshed by ensuring a bilateral ceasefire," a Kremlin statement said.
Saltanov also thanked Israel for allowing Russia to evacuate its nationals and those of former Soviet republics from Gaza amid the violence.
Two flights carrying 102 Russians and 76 nationals from Ukraine, Kazakhstan, Belarus and Moldavia evacuated from Gaza arrived in Moscow earlier on Saturday. Some 250 Russians, mainly the wives and children of Palestinians who studied at Russian universities, were reported to be living in the Gaza Strip.
Kazakhstan's Foreign Ministry said that some 14 Kazakh women, the wives of Palestinians, had chosen to remain in Gaza.
