Moscow's appeal came after US Secretary of State Rex Tillerson encouraged Saudi Arabia and its allies to ease their land and sea "blockade" of Qatar.
On Monday, Saudi Arabia, Bahrain, Egypt, the United Arab Emirates, Yemen and the Maldives severed ties with Qatar, accusing it of backing terrorism. Qatar called the accusations baseless.
On Saturday HE the Foreign Minister Sheikh Mohammed bin Abdulrahman al-Thani was in Moscow to see Russian Foreign Minister Sergei Lavrov, a day after visiting Germany and Brussels.
"We cannot be happy in a situation when the relations between our partners are worsening," Lavrov said.
"We are in favour of resolving any disagreements through... dialogue."
Russia is "ready to try to do everything in its power" to help resolve the crisis, he said.
HE the Foreign Minister said his aim was to inform Russia about "the illegal measures" taken against Qatar.
"Differences are always solved by dialogue and the (Gulf) Cooperation Council is the most suitable framework for these talks," he said.
'Blockade hindering military'
Trump's Qatar comments Friday overshadowed earlier remarks by Tillerson that the stand-off was hindering the US-led fight against the Islamic State group.
"The blockade is hindering US military actions in the region and the campaign against ISIS," Tillerson warned, referring to the land and sea blockade imposed on Qatar.
Pentagon spokesman Captain Jeff Davis said the "evolving situation is hindering our ability to plan for longer-term military operations".
Tillerson said it was also having humanitarian consequences.
Human cost
Amnesty International echoed him, saying the sea and land blockade and other "drastic" measures against Qatar were taking their toll on families, workers and students.
"Saudi Arabia, Bahrain and the United Arab Emirates are toying with the lives of thousands of Gulf residents as part of their dispute with Qatar, splitting up families and destroying people's livelihoods and education," the London-based watchdog said.
It noted that Bahrain, Saudi Arabia and the UAE had warned of harsh punishment, including up to 15 years in jail, if people "dare to criticise these measures" against Qatar.
The measures include banning Qatar Airways from airspace and closing Qatar's only land border with Saudi Arabia. The Arab states have also ordered Qataris out within 14 days.
