Chinese Foreign Minister Wang Yi will attend the meeting that follows weeks of warnings from the US administration that it will no longer tolerate North Korea's missile launches and nuclear tests.
The United States has called for stronger UN sanctions on North Korea, but it wants China to take the lead in diplomatic efforts to resolve the crisis.
"We are going to be discussing what steps may be necessary to increase pressure on Pyongyang to have them reconsider their current posture," Tillerson said in an interview to Fox News on the eve of the meeting.
Turning to China, Tillerson said: "We are going to test their willingness to help us address the serious threat."
No resolutions will be adopted at the meeting, but it will allow the United States and its allies to put the onus on China to use its leverage to rein in Pyongyang.
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North Korea is seeking to develop a long-range missile capable of hitting the US mainland with a nuclear warhead, and has so far staged five nuclear tests, two of them last year.
China, Pyongyang's number one trade partner, has repeatedly called for a return to talks on denuclearization but has been reluctant to use economic pressure that could destabilise North Korea.
The council has imposed six sets of sanctions on North Korea -- two of which were adopted last year -- to significantly ramp up pressure and deny Kim Jong-Un's regime the hard currency revenue needed for his military programs.
But UN sanctions experts have repeatedly told the council that the measures have had little impact on Pyongyang because they have been poorly implemented.
The United States, which holds the council presidency this month, will urge UN member-states to take steps to fully implement the sanctions, the toughest currently to hit any country.
WATCH: UN must ready new sanctions on North Korea: Trump
Bringing Kim Jong-un to his senses
US President Donald Trump has made North Korea his top foreign policy priority, telling Security Council ambassadors at the White House this week that he was determined to address the crisis head-on.
"People have put blindfolds on for decades, and now it's time to solve the problem," said Trump.
The meeting of the top UN body comes just days after South Korea received the first deliveries of equipment for a new missile defence system from the United States.
Washington and Seoul are deploying the Terminal High Altitude Area Defense (THAAD) which is meant to intercept and destroy short and medium-range ballistic missiles during their final phase of flight.
China has fiercely opposed the deployment of the THAAD system and warned that it will stoke tensions on the peninsula.
Beijing has proposed a freeze on North Korea's nuclear and missile activities in exchange for a halt to the annual US-South Korea military drills that have infuriated Pyongyang.
That offer has been flat-out rejected by the United States, which said the onus was on North Korea to show that it was willing to open up for talks.
As the United States presses on with its new diplomatic strategy on North Korea, it has also deployed an aircraft carrier strike group led by the USS Carl Vinson to the western Pacific.
US officials have said that while military action is an option, a diplomatic solution was the preferred outcome.
"We want to bring Kim Jong-Un to his senses, not to his knees," said US Admiral Harry Harris, who heads Pacific command.
