The nations also said a military intervention "has not been and will not be considered" to end the crisis, something mentioned by Sheikh Sabah in his remarks.
President Trump speaks with Middle East leaders as a diplomatic rift between several gulf states continues.
In a phone conversation on August 30, Trump told the Saudi monarch that a diplomatic resolution was necessary in order to fulfill a commitment Washington and its regional allies had made to stay united while fighting terrorism.
Kuwait has been acting as a mediator since June 5, when Saudi Arabia, the UAE, Bahrain and Egypt announced they were severing relations with Qatar and imposing a land, sea and air blockade against it, accusing Doha of supporting "terrorism". "But it is something that could happen", Trump said. "Kuwait has been really the leader of getting it solved, and we appreciate that very much".
Qatar, which is a US ally and hosts a large American military facility, has denied the allegations.
"Nothing is inevitable. It would be great if something else could be worked out", said the president.
18 soldiers killed in ISIL ambush in Egypt's Sinai
The assailants, whose identities were not yet known, later opened fire at survivors and commandeered a police pickup truck. Police said they killed 10 extremists in two simultaneous raids on apartments in a densely populated Cairo neighborhood.
Kuwait has been trying to mediate the dispute between Qatar and its US-allied Arab neighbors. The U.S.is also concerned that the spat among Arab governments could hamper a shared interest in blunting Iranian assertiveness in the region.
The Kuwaiti leader is thanking the United States for standing with their country when Iraq's Saddam Hussein launched an invasion.
Trump told a news conference on Thursday that he would be willing to step in and mediate a dispute among the US -allied Arab states and Qatar, and said he thought a deal could come quickly.
President Donald Trump on Thursday hailed efforts by the leader of Kuwait, a staunch American ally, to mediate a festering diplomatic crisis involving Qatar and its Arab neighbors that could have implications for the US military presence in the region.
Shaikh Sabah - who is accompanied by a high-calibre official and business delegation - and President Trump will meet at the White House on Thursday. He initially appeared to side with Saudi Arabia.
He emphasized, "what is important is that we have stopped any military action". Some 6,000 North Koreans work in Kuwait.