White House officials have talked about getting tough with North Korea because of its expanding nuclear program, and China is helping to keep North Korea afloat.
"President Trump indicated to President Xi that he welcomed any ideas that President Xi and China might have as to other actions we could take and that we would be happy to work with them", Tillerson said.
Trump has received widespread support for the military action, taken to punish Syrian President Bashar Assad's government after the US concluded he used chemical weapons against Syrian civilians.
China has sided with Russian Federation at the United Nations in opposing condemnation of Assad's government but has not become directly involved in the conflict. Spokeswoman Hua Chunying told reporters that Chinese officials "oppose the use of chemical weapons by any country, organization or person for any objective and under any circumstance" and that China "supports relevant United Nations agencies in carrying out independent and comprehensive investigations into all uses or suspected uses of chemical weapons and, on the basis of solid evidence, reaching a conclusion which can stand the test of history and facts". The move sent the important message that the U.S.is still the "preeminent power", according to a seasoned diplomat and China hand.
North Korea has said the USA missile strikes on Syria were an "intolerable act of aggression" that "proves a million times over" that strengthening its nuclear programme was the right choice, state media reported Saturday. White House senior adviser Jared Kushner is seated at left.
Diplomatically isolated North Korea considers Syria a key ally.
In fact, the North justifies its nuclear weapons as a necessary deterrent to the US military threat and has over the past weeks warned that the biggest-ever joint U.S.
U.S. Commerce Secretary Wilbur Ross said the Chinese had expressed an interest in reducing China's trade surplus as a way of controlling their own inflation.
"But we have developed a friendship - I can see that - and I think long term we are going to have a very, very great relationship and I look very much forward to it".
However, Xi apparently got something out of dinner.
"President Trump noted the challenges caused by Chinese government intervention in its economy and raised serious concerns", Tillerson said.
Trade remains Trump's chief grouse against China, which has a trade surplus of $347 billion with the USA, and he, as presidential candidate, had accused China of cheating, by manipulating their currency, and had once even accused China of "raping" the US.
He also last week signed a pair of executive orders focused on reducing the US trade deficit, an apparent shot at China, which accounted for the vast bulk - $347 billion - of last year's $502 billion trade deficit.
"Both the atmosphere and the chemistry between the two leaders was positive... all of us are feeling very good about the results of this summit", said Tillerson.