
"Our goal is to encourage the international community to start cooperation with the Palestinian government and pressurize Israel to recognize the Palestinian state's right to exist," RIA Novosti news agency quoted him as saying.
"We value Russia's position towards lifting the blockade from which the Palestinian people suffer. We also value Russia's special position in the issues of the Middle East settlement."
Russia takes a softer line on the Islamist Hamas than the United States and the European Union, which along with the United Nations make up the Quartet.
The West is boycotting Hamas and the government it formed after an election victory last year, demanding it recognizes Israel, renounces violence and accepts interim peace deals.
Earlier this month, Hamas and the rival Fatah faction agreed to form a unity government hoping to end the international boycott and factional fighting.
But the Quartet failed to work out a common stand on the unity government at a meeting last week.
Russia wants the Quartet to back the coalition deal, while the Western governments have adopted a wait-and-see approach.
Moscow is the last leg of Mashaal's tour, which included Egypt and Iran.
He is to hold meetings in the Russian foreign ministry, but there were no other details available. It was unclear if he would meet Russian President Vladimir Putin. It is believed he will be in Moscow for at least two days.