The king said the world's two top oil exporters would continue to collaborate to stabilise oil markets. On Wednesday Putin said an extension of the OPEC deal was possible and could last "at least to the end of 2018".
During the visit King Salaman said he was looking forward to building a strong relationship with Russian Federation "in the interests of peace, security and development of the world economy".
Russian Federation would like Saudi Arabia to become a shareholder in the project but the source said there were also other roles the kingdom could take. The two nations - joint architects of the supply agreement between the Organization of Petroleum Exporting Countries and other producers that has boosted prices - are looking at ways to extend and deepen their relationship.
The statement came as Saudi Arabia's King Salman visited Moscow.
Earlier in the day, King Salman met with the Russian leader to discuss a number of issues, including the oil market situation, defense industry cooperation, situation in Syria and the diplomatic rift between a number of Arab states and Qatar.
"Because by law, I think the presidential election campaign will be officially announced at the end of November, beginning of December", Putin said at the Russian Energy Week Forum.
Russian Foreign Minister Sergei Lavrov in an interview has said that the visit of King Salman would contribute to stability in the Middle East.
The two have profound disagreements on Syria - where more than 330,000 people have been killed since war erupted in 2011 - and Yemen, where a Saudi-led coalition has been bombing Houthi rebels since 2015, drawing criticism from Moscow.
Additionally, Russia's state atomic corporation Rosatom and the Saudi Energy Ministry signed a nuclear energy cooperation accord.
Russian Economy Minister Maxim Oreshkin said on October 3 investors were interested in a nuclear energy plant Saudi Arabia wants to build and Saudis would be offered investment opportunities including in shipping company Sovcomflot.
Despite this, other factors weighed on oil prices, including the return to production of Libya's Sharara oilfield after an armed brigade forced a two-day shutdown.
The head of Saudi's state-owned oil giant Aramco Amin Nasser told Rossiya 24 state television ahead of the talks that the company would sign agreements with Russia's Gazprom, Gazprom Neft and Sibur energy companies as well as with an affiliate of Lukoil.