Tillerson will deliver this message on his first official trip to Asia, a three-country tour that promises to be a tightrope walk of diplomatic tensions -- and nowhere more so than in Beijing, North Korea's closest ally and protector.
While Tillerson will tell Beijing that the US is tired of Chinese companies helping facilitate Pyongyang's weapons program, he'll also seek to engage Chinese support for a broader attempt to rein in North Korea -- resembling the international coalition that created the Iran nuclear deal. He'll do this as he lays the groundwork for Chinese President Xi Jinping's visit to the US in April.
Tillerson will also engage on the usual areas of US-Chinese tension, including the country's claims to contested waters in the South China Sea; trade; the status of Taiwan; and the recent US deployment of a defensive missile system and drones to South Korea.
But none of these issues are as fraught or potentially dangerous as the situation with Pyongyang, which seems to have accelerated its nuclear program in the last few months with the aim of perfecting a nuclear weapon and the means to deliver it as far as the US.
"China has been part of the problem rather than part of the solution," said Bruce Klingner, a senior research fellow at the Heritage Foundation, who describes North Korea as a global military threat.
China has "acted like North Korea's lawyer at the UN Security Council. They deny evidence of North Korea wrongdoing, they insist on loopholes, they insist on watering down what would otherwise be more effective resolutions" at the United Nations, Klingner said.
"And whenever North Korea does some kind of a provocation or violation," he continued, "they have this value-neutral response of calling on both Koreas not to raise tensions, when it is only their Korea which is doing so."
Acting State Department spokesman Mark Toner said discussions between Tillerson and his Chinese counterpart about dealing with North Korea have already been taking place. They are looking at "what are next steps we can take to really put pressure on the regime to make them feel and pay a price for their behavior," he said.
He acknowledged that the US wants to see China do more.
"We're always cognizant of China's influence over North Korea, and we're always encouraging it to play a more forceful role in that regard, whether it's through a more thorough implementation of the sanctions regime that exists, or through other ways," Toner said Monday. "So that's a leverage that China brings to the table, and that's certainly something we want to see them take more advantage of."
In an rare interview with CNN Thursday, an official with China's Ministry of Foreign Affairs said the Chinese hoped the US would not move forward with the new penalties.
"We hope they will not," said Xiao Qian, the Director General of the Asian Affairs department. "Because then it is not fair and that's not right. That's not the correct way of dealing with things."
When pressed on how the Chinese would respond if the US moved forward with new penalties on its banks and businesses, Xiao avoided a direct answer, saying only that the Chinese would wait to hear directly from the State Department.
Given that China represents 90% of North Korea's economic activity, Beijing has leverage, Klingner said. Chinese firms that do business with the isolated country can operate as front companies that allow it to sell weaponry and other items overseas, and import goods.
US officials said that the administration is considering targeting Chinese firms with sanctions for economic engagement with North Korea, given evidence that some of them are helping the country import and export weapons.
"We can influence the Chinese banks and businesses that are dealing with North Korea. It has worked in the past," he said.
He noted that in the mid-2000s, the Bank of China defied the Chinese government and cut off its interaction with North Korea to avoid the risk of US sanctions.
"So regardless of what Beijing wants," Klingner said, "we can influence the behavior of those Chinese entities that are engaging in economic activity with North Korea."
The Trump administration has already dipped a toe into sanctioning Chinese companies and by doing so is looking to expand a policy that the Obama administration began last year.
The Commerce Department last week announced that a Chinese tech firm, ZTE, would pay a $1.2 billion fine for violating sanctions by selling equipment to Iran and North Korea.
In September, the Obama administration targeted a Chinese company called the Dandong Hongxiang Industrial Development Co. Ltd for ties to North Korea -- the first time the Obama administration hit a Chinese firm with sanctions for dealing with North Korea and supporting its nuclear program. On the same day, the Justice Department unsealed four indictments for money laundering by te same firm and four individuals on behalf of Pyongyang.