Tillerson is also scheduled to meet with his British counterpart and a French foreign ministry official in London later on Thursday.
The US hopes the trip will give "new energy and focus to mediation efforts led by the United Nations", Hook said.
But in the face of resistance from China and Russian Federation, the U.N. Security Council on Monday imposed only restrictions on the North's oil imports, among other sanctions.
"Britain will continue to work closely alongside our partners to tackle the shared challenges we face worldwide".
"United Nations Security Council resolutions, including the most recent unanimous sanctions resolution, represent the floor, not the ceiling, of the actions we should take", he said in the statement on Friday.
Mr Tillerson visited No 10 for detailed talks with the UK National Security Adviser Mark Sedwill today before a broader meeting with Foreign Secretary Boris Johnson. Merkel suggested that the approach taken with Iran - lifting global sanctions in exchange for it curbing nuclear activities - could be used with North Korea.
On the Iran nuclear deal, which is viewed sceptically in Washington, May was "underlining its importance in preventing Iran from procuring nuclear weapons", the spokesman said.
"These continued provocations only deepen North Korea's diplomatic and economic isolation", the secretary of state added.
Tillerson said the UN Security Council's sanctions against North Korea, including the most recent ones adopted on Monday, represent "the floor, not the ceiling" of the actions the global community should take.
Nicolas de Riviere, the head of political affairs in the French Foreign Ministry, also joined the USA and British officials to discuss hurricane relief work in the Caribbean.
The Foreign Secretary, who had just returned from a trip to see the devastation in the Caribbean, set out the UK's efforts to support its overseas territories.
The meeting is meant to try and find a way to break the political deadlock in Libya and bring stability to the North African country in a bid to help deal with the issue of illegal migration.