The Palestinian leader was set to visit Moscow two weeks after a visit by Israeli Prime Minister Benjamin Netanyahu.
Abbas has refused any contact with US President Donald Trump's administration since Washington's recognition of Jerusalem as the Israeli capital at the end of last year. Abbas is due to speak at the United Nations Security Council on February 20.
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Palestinians see the US decision, which broke with years of international diplomacy, as a denial of their claim to East Jerusalem as the capital of an eventual Palestinian state.
Israel took control of East Jerusalem in the 1967 Six-Day War, annexed it and later declared it the indivisible capital of Israel.
The US Ambassador to the UN, Nikki Haley, has accused Abbas of lacking the courage needed to forge a peace deal with Israel.
Abbas in turn has rejected any mediation by Washington in the Israeli-Palestinian conflict and has promised his people to work towards full recognition of a Palestinian state by the United Nations.
Alexander Shumilin, a Middle East scholar at the Institute for US and Canadian Studies, called Abbas's visit "an attempt to cosy up to Russia, a consistent ally, and to stop Netanyahu leading Moscow astray during an improvement in Russia-Israeli ties".
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Netanyahu visited Russia on January 29 and along with Putin attended a memorial ceremony at the Jewish museum in Moscow for the victims of Nazi camps. He took the opportunity to accuse Iran of wanting to "destroy" the Jewish state.
In turn, the Russian president likened antisemitism to "Russophobia" and said Russia and Israel were "cooperating closely", particularly against "attempts to falsify history".
For Shumilin, Monday's visit "is a necessary political gesture for Abbas but can do little in the practical sense".
"It is also definitely not worth expecting a breakthrough from this visit," he added.
In 2016 Russia offered to host one-on-one talks without preconditions between Abbas and Netanyahu but these never materialised.
Moscow 'understands' Palestinian anger at Trump: Lavrov
In January, Russian Foreign Minister Sergei Lavrov estimated that chances of resuming direct talks between the two sides in the current situation were "close to zero".
Lavrov also said "we understand the emotions" Palestinians feel towards Trump.
"We keep hearing in recent months that the US is about to publish some 'major deal' that... will satisfy everyone," he said. But he added that Russia "has not seen or heard of such a document or even any statement".
With relations between Washington and Moscow at a record low for the post-Cold War era, Abbas may be expecting that "Russia-US relations will get even worse and then Russia could do something to spite the US", Shumilin said.
On November 29, 2012, the United Nations designated Palestine as a non-member observer state after a vote by the General Assembly.
That enabled the Palestinians to join international organisations and the International Criminal Court, though they did not become a full UN member state.
Palestinian statehood is recognised by more than 130 countries.
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