Saudi Arabia and Russia are "very, very close" to a deal on oil production cuts, Kirill Dmitriev, head of Russia's sovereign wealth fund, told CNBC on Monday.

"I think the whole market understands that this deal is important and it will bring lots of stability, so much important stability to the market, and we are very close," Dmitriev, who is also one of Moscow's top negotiators, told CNBC.

Dmitriev was first to make a public declaration of the need for an enlarged supply pact, potentially involving producers outside the OPEC+ group that currently consists of the Organization of Petroleum Exporting Countries (OPEC) and some other oil producers led by Moscow.

A previous three-year deal to stabilize oil prices collapsed a month ago, with Saudi Arabia and Russia blaming each other for failure to find a compromise at an OPEC+ meeting in Vienna on March 6.

Saudi Arabia and Russia were initially set to meet on Monday to discuss output cuts, but that has now been pushed back to April 9, as oil prices continue to come under pressure.

G20 energy ministers and members of some other international organizations will hold a video conference to be hosted by Saudi Arabia on April 10, a senior Russian source told Reuters on Monday, as part of the efforts to get the United States involved in a new deal on production cuts.

The two countries have been locked in a price war since last month, pushing down the oil price to two decades lows. 

The chief executive of Russia's VTB Bank, a bank that mainly service Russia's foreign trade also said that Russia is very interested in stabilizing oil prices, Andrey Kostin told CNBC.

"No one is interested in low oil prices. Neither the United States nor Russia, nor the Saudis," he said. 

"From this point of view, I think there should be a reasonable agreement achieved at the end of the day," he told CNBC.