Duterte was unable to attend the Gala dinner hosted by President Tran Dai Quang last Friday. He added that he is "a very good mediator and arbitrator".
Duterte won last year's presidential elections after promising to eradicate illegal drugs with an unprecedented crackdown that would see up to 100,000 people killed.
"If you do not like me, I do not like you".
China is claimant to swathes of ocean, extending far outside its internationally agreed 200-mile economic exclusion zone (EEZ).
The two met on the sidelines of the Asia-Pacific Economic Cooperation (APEC) summit in the Vietnamese resort city of Danang.
The controversial issue was also understood to have figured in a meeting of officials from India, the US, Japan, and Australia under the proposed Quadrilateral coalition of the four countries.
"You want to control the passage, or do we have free passage?"
Hours after being sworn in as Philippine president, Duterte goes to a Manila slum and urges residents to kill drug-addled neighbours.
The South China Sea dispute will be one of the major issues at the East Asia Summit in The Philippines this week. He had bilateral meetings with Putin and Chinese President Xi Jinping.
China and Vietnam need to "stay committed to seeking a fundamental and durable solution to their maritime issues acceptable to both sides through friendly consultation", Xi said in an article published by the state-run Xinhua news agency Thursday - before Xi's arrival in Vietnam.
China, Taiwan, Vietnam, the Philippines, Malaysia and Brunei all have competing claims to various island chains in the South China Sea.
Trump's Asian tour comes as the United States rallies for tougher sanctions against North Korea's nuclear program and ballistic missile tests.
"While the South Korean people are opposing war threats against the north and demanding peace in the region, here is the president of the Philippines who could not even commit to peace in his own country, allowing the country as a launching pad of U.S. aggression and putting the Filipino population to the danger of nuclear obliteration", he said in a statement.
In recent days, North Korean state-run media has taken to referring to Mr Trump's age.
Various sectoral groups staged a protest near the US embassy here on November 10, the second day in a row, in anticipation of the first visit by US President Donald Trump in time for the Asean summit on November 12.
"(US) Foreign policy in Asia has had its oxygen sucked out the room", he said.
Leaders at the Philippines summit will discuss the South China Sea, but mainly to agree on a procedural step to cool tensions. "For China, this is their time".