By Gabriela Baczynska

MOSCOW, Dec 18 (Reuters) - Russia may ease restrictions on the Muslim Brotherhood soon to improve relations with Egypt and rebuild influence lost during the Arab Spring revolutions, diplomatic sources say.

The election of President Mohammed Mursi, propelled to power by the Islamist group, offers President Vladimir Putin a chance to improve relations with Cairo that were strained during the long rule of Hosni Mubarak, who was ousted in 2011.

Russia's Supreme Court banned the Muslim Brotherhood from operating in Russia in 2003, describing it as a terrorist organisation.

But Moscow is now trying to beef up ties with Egypt, partly to offset some of the influence it has lost in the Arab world in the past two years, particularly in countries such as Libya and Syria that have been recipients of Russian arms.

Easing restrictions on the Muslim Brotherhood would follow

a similar move by the United States, which tweaked its ban on formal contacts with the Islamist group, banned under Mubarak, early in 2012. Mursi is expected to visit Washington in 2013 for the first time since his election in June.

Western diplomatic sources say Mursi also accepted an invitation to visit Russia when Foreign Minister Sergei Lavrov visited Egypt during a Middle East tour last month.

"The visit is expected close to the end of the first quarter in 2013. The Brotherhood being on the list remains the problem and Lavrov is said to have given assurances that they will deal with that," one of the sources said.

Russia has in the past accused the Muslim Brotherhood of supporting rebels who want to create an Islamist state in Russia's mainly Muslim North Caucasus.

The Kremlin is still struggling to contain the Islamist insurgency, which Putin has warned could fuel violence in other regions closer to the capital.

But during his trip to Cairo, Lavrov endorsed an initiative by Mursi to resolve the conflict in Syria and political analysts say Moscow appears to be looking for ways to engage more with Egypt, a popular holiday destination for Russians as well as being a regional power.

BETTER TIES?

"As far as I know, Minister Lavrov wants to 'delist' it (the Muslim Brotherhood)," said Alexei Grishin, head of the Religion & Society think tank who used to be a presidential adviser on Islam.

"Any fresh decision by the Supreme Court would be a very lengthy procedure, so maybe what can be done is to restrict the blacklisted Muslim Brotherhood only to the faction that fought in Russia, in the Caucasus, for example," he told Reuters.

Russia's Supreme Court and anti-terrorist committee were not available for comment. The Foreign Ministry declined immediate comment.

The Arab world's biggest nation is engulfed in a prolonged political crisis nearly two years after the fall of Mubarak that pits Mursi's Islamist supporters against liberal, secular and Christian opposition groups.

"Nobody really knows what will come out of this, but Egypt is such an important country that no matter what we think of its new authorities, we need to have a dialogue," said Fyodor Lukyanov, a prominent foreign policy analyst.

"It makes sense for Russia to make a bet on Egypt, or at least seek ties with them very actively. So I think they will soon solve the issue with the Muslim Brotherhood in Russia."

In Egypt, a Muslim Brotherhood spokesman said the ban on the group in Russia was "a very bad thing" and made clear any real improvement in ties was unlikely before that was settled.

"Our leader (the head of the Brotherhood group) had mentioned that issue to the Russian ambassador in Cairo during a joint meeting where he told the ambassador: 'How come you are asking to have a strong relationship with us while you see as a terrorist group?'," spokesman Mahmoud Ghozlan said.