Hamas defeated the mainstream Fatah movement in a surprise landslide victory in Palestinian polls on 25 January. It is expected to form a new government soon.
Vladimir Putin made the invitation while in the Spanish capital, Madrid.
"We are maintaining our contacts with Hamas and intend, in the near future, to invite the leadership of this organisation to Moscow," he said.
The US and the European Union have called on the resistance group to renounce violence and disarm its fighters. Hamas, considered a terrorist organisation by the US, does not recognise Israel.
Putin went on to say that he did not believe in burning bridges.
"We have never called Hamas a terrorist organisation," he said.
Putin said: "It has to be recognised that Hamas came to power in the Palestine autonomy through a democratic and legitimate election and one should respect the choice of the Palestinian people.
"But ... we must also seek steps that would be acceptable both for the political forces leading the Palestinian autonomy, for the international community and for Israel," he said.
"We are deeply convinced that burning bridges, especially in politics, is the easiest thing to do but it has little future."
Mashaal: Qatar is not mediating
between Hamas and the US
Alexander Kalugin, Russia's special envoy for the Middle East, said later from Moscow that Russia would ask Hamas to recognise Israel's right to exist.
"There cannot be any dialogue without it," he was quoted as saying by the RIA-Novosti Russian agency.
Kalugin said that if Hamas representatives were to come to Russia, the meeting would be in line with agreements reached by the so-called Quartet, a group of Middle East peace negotiators which includes the US, Russia, the EU and the UN.
Ismail Haniya, a senior Hamas official, said after Putin made his comments: "If we receive an official invitation to visit Russia, we will visit Russia."
Mark Regev, the Israeli Foreign Ministry spokesman, said Israel would not negotiate with Hamas until it "recognises Israel's right to exist, renounces terror and accepts the Middle East peace process".
An Israeli government source said: "People in Jerusalem are raising an eyebrow - what's going on here?"
On Thursday, Khaled Mashaal, Hamas' political leader, denied Qatar was mediating between his group and Washington.
Mashaal was speaking in Doha while on a tour of Arab states designed to shore up support for his movement. Earlier this week he was in Egypt
After his meeting with the Qatari Amir Sheikh Hamad bin Khalifa Al-Thani, Mashaal reaffirmed the need to form a national coalition government and emphasised the need for providing assistance for the Palestinians in light of threats by Western nations to suspend international aid.
Zahar reaffirmed Hamas would
not prevent attacks on Israel
Mashaal urged the US and Europe on Thursday not to cut off funds to the Palestinian Authority, while saying Hamas will neither renounce violence nor recognise Israel.
Mashaal said: "Hamas can never trade its principles for money."
He said Hamas would not compromise its fundamental principles to secure Western finance. "We are capable of finding alternatives" to Western aid, he said.
"We have never called Hamas a terrorist organisation"
These would include money saved by eliminating corruption and receiving aid from Arab countries, which have made "good promises and commitments", Mashaal said.
Separately, Mahmoud Zahar, who leads Hamas inside the Palestinian territories, said on Thursday that a Hamas government would not stop others from attacking Israel.
Speaking a day after the Islamic Jihad group declared that it would continue its attacks, Zahar told Aljazeera: "We do not forbid anyone to resist because we are, and will continue to be, a programme of resistance."