However, both men struck a friendlier tone about broader cooperation between the United States and Russian Federation after what they described as "frank" preliminary talks in the Black Sea resort of Sochi.
Iran was another critical agenda item amid spiraling tension between Washington and Tehran that some fear could lead to conflict.
But he added: "We have also made clear to the Iranians that if American interests are attacked, we will most certainly respond in an appropriate fashion".
His inquiry had cast a pall over U.S.
On Tuesday, Saudi Araba said drones attacked one of its oil pipelines as other assaults targeted energy infrastructure elsewhere in the kingdom.
Pompeo - who like most of the USA political class, with the major exception of President Donald Trump, has been outspoken in his criticism of Putin - said that the two countries found areas on which to cooperate, and voiced optimism in breaking a deadlock on Syria.
But Pompeo told reporters Tuesday he thinks US and Russian officials "now can begin to work together" in a way to begin the process set out in a U.N. Security Council resolution that calls for a Syrian-led political transition with a cease-fire, new constitution and elections.
US Secretary of State Mike Pompeo met with Putin and Russian Foreign Minister Sergei Lavrov on Tuesday.
Russian Federation and the United States voiced hope Tuesday for better ties including working together in Syria as President Vladimir Putin welcomed Secretary of State Mike Pompeo, but tensions were laid bare in a clash over election meddling.
Putin said he had spoken to U.S. President Donald Trump several days ago, and he got the impression that Trump indeed "intends to rebuild U.S". "For our part, we have repeatedly said that we would also like to restore full-format relations".
Pompeo's visit represents the first high-level contact between Moscow and Washington since U.S. Special Counsel Robert Muller submitted a report examining the nature of Russia's role in the 2016 U.S. presidential election.
"We of course heard President Trump's statement that he expects to hold a meeting with President Putin, " Lavrov said in the news conference.
A day after Trump said he would meet with Putin at the Group of 20 summit of leaders in Japan in June, Lavrov said Russian Federation has not received a formal proposal. These included Iran, Syria and Venezuela, where the United States supports opposition leader Juan Guaido as interim president and Russian Federation backs President Nicolas Maduro.
Putin told Pompeo his recent telephone conversation with Trump raised hopes for an improvement in relations. "It's not destined that we're adversaries on every issue". Putin did not address Mueller's conclusion that Russian Federation actively interfered in the election. -Russian efforts to bring a United Nations -brokered end to the war that began in 2011 fell apart in 2016. At the time, Trump cited Russia's failure to release Ukrainian ships and sailors seized during clashes in the Kerch Strait near Crimea as the reason for his refusal to meet Putin.
"We regret that the deal is falling apart", Putin said following talks with Austrian President Alexander van der Bellen on Wednesday.