WASHINGTON (Reuters) - U.S. Secretary of State Rex Tillerson said Washington was considering imposing new sanctions on North Korea if it takes steps that deserve a further response, and is threatening to impose secondary sanctions on those doing illicit business with Pyongyang.

“We are preparing additional sanctions if it turns out North Korea’s actions warrant additional sanctions,” Tillerson said on Wednesday in remarks to State Department employees.

The vast majority of North Korea’s trade is with its key ally China, and so any hard-hitting secondary sanctions would likely target Chinese firms.

Tillerson called on countries all over the world to implement existing U.N. sanctions on North Korea’s nuclear weapons and missile programs, adding that the U.S. administration would be willing to use secondary sanctions to target foreign companies that continue to do business with Pyongyang.

“We told them we’re watching what you’re doing, when we see you not implementing, we see companies, we see individuals that are violating these sanctions, we’re going to contact you, we’re going to ask you to take care of it,” Tillerson said.

“If you can’t take care of it, or you simply don’t want to take care of it for your own internal political reasons, we will. We’ll sanction them through third-country sanctions.”

North Korea has stepped up its weapons tests, firing dozens of missiles and detonating two nuclear bombs since the start of last year. The most recent missile test, which failed, came on Friday following a U.N. Security Council meeting chaired by Tillerson at which he urged the council to act before North Korea did.

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Tillerson said the Trump administration had been “leaning hard into China” ... “to test their willingness to use their influence, their engagement with the regime.”

The Obama administration began blacklisting Chinese firms doing illicit business with North Korea, but such steps have so far been limited.

Diplomats said this week Washington was negotiating with China on a possible stronger U.N. Security Council response - such as new sanctions - to North Korea’s missile launches.

Chinese Foreign Ministry spokesman Geng Shuang said on Wednesday U.N. resolutions were clear that further measures would be taken in the event of more nuclear or missile tests.

The last round of complex sanctions imposed by the Security Council took three months to negotiate following Pyongyang’s fifth nuclear test in September. Those aimed to cut North Korea’s annual export revenue by a quarter.

North Korea’s state media published a rare, strong, criticism of China on Wednesday, saying Chinese state media commentaries calling for tougher sanctions were undermining relations.