In a phone conversation held on the Saudi initiative, the two leaders "agreed to deepen coordination between Russia and Saudi Arabia on [issues of] regional and international agenda," the press service said in a statement.
The Russian president invited the Saudi monarch to pay a visit to Russia, the Kremlin added without giving any further details.
Both leaders "expressed bilateral determination to boost mutually beneficial bilateral cooperation in different areas," it said.
According to the statement, King Salman praised Russia's efforts to settle the Yemen crisis, including its backing of the April 15 UN resolution that banned arms supplies to Houthi rebels in Yemen.
Yemen has been in turmoil since the Houthi militant group took over large swathes of land in the country, prompting President Abd Rabbuh Mansur Hadi to allegedly flee to Riyadh. Since March 2015, a coalition headed by Saudi Arabia has been bombing rebel positions at the request of Hadi.