"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, Reuters wrote.
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 said on Monday, as part of the efforts to get the United States involved in a new deal on production cuts.
Oil prices fell on Monday after Saudi Arabia and Russia delayed the meeting as the coronavirus pandemic pummels demand.
Brent crude fell more than $3 when Asian markets opened but recovered some ground, with traders hopeful a deal between the top producers was still within reach.
Brent was down $1.08, or 3.2%, at $33.03 a barrel. US crude was $1.49, or 5.3%, lower at $26.85 a barrel, off a session low of $25.28.
US Energy Secretary Dan Brouillette said that after speaking with the energy ministers of Saudi Arabia and Russia he hoped the countries will end their war over market share this week.
Kremlin spokesman Dmitry Peskov also said Moscow was ready to coordinate with other oil exporting countries to help stabilize the market and that the OPEC+ meeting was delayed for technical reasons.
OPEC+ is working on a deal to cut production by about 10% of world supply, or 10 million barrels per day (bpd), in what member states expect to be an unprecedented global effort.
Monday's slump in crude benchmarks came after a strong surge on April 2 and 3, with Brent gaining 38% and WTI 39.5% over the two days as the market reacted to US President Donald Trump's tweets that a deal to cut output was imminent.
Investor morale in the euro zone fell to an all-time low in April and the bloc's economy is in deep recession because of the novel coronavirus, a survey showed on Monday.
Markets were also spooked when the National Health Commission of China said on Monday that 78 new asymptomatic cases had been identified as of the end of the day on Sunday, compared with 47 the day before.
Asymptomatic patients, who show no symptoms but can still pass the virus to others, have become China's chief concern after strict containment measures succeeded in cutting the overall infection rate.