The request was made during a just-concluded visit to Moscow by a delegation from the Federal National Council, a part-elected assembly which advises the government, said FNC speaker Abdul Aziz al-Ghurair.
"At the request of the FNC team, the Russian foreign ministry voiced readiness to mediate to convince Tehran, which has good relations with Moscow, to recognise that there is a problem between the UAE and Iran over the three islands," the official WAM news agency quoted Ghurair as saying.
Ghurair, who led the delegation that returned home on Friday, said Sergey Mironov, chairman of Russia's Federation Council, the upper house of parliament, backed the UAE's claim to the strategic islands.
The oil-rich UAE, supported by other Arab states, has repeatedly proposed resolving the dispute through direct negotiations or international arbitration, but Iran has always refused.
Iran, then ruled by the pro-Western shah, gained control of Greater Tunb, Lesser Tunb and Abu Musa as British forces left the Gulf in 1971.
Iran took possession of Greater and Lesser Tunbs, while the only inhabited island — Abu Musa — was placed under joint administration under a deal with Sharjah, now part of the UAE.
But since then, the UAE says, the Iranians have taken control of all access to the island, installed an airport and military base there, as well as encouraged settlers to move in to change its demographic make-up.
The three islands lie roughly half-way between the Iranian port of Bandar-e Langeh and Dubai in the UAE.
The dispute has been a sore point in relations between the UAE and Iran despite wide-ranging links. The UAE is Tehran's top trade partner and an estimated 450,000 Iranians live in the country.
