Sheikh Mohamed told reporters in Ankara that Qatar would continue to work with the US and Kuwait to end the rift with four Arab countries, according to Al Jazeera.
Urging Saudi Arabia, the UAE, Bahrain and Egypt to engage in talks, he said Qatar wanted dialogue based "on mutual respect for state sovereignty and respect for international law".
The four countries severed ties with Doha on June 5.
Tillerson concluded his mediation efforts on Thursday, making no promise of an imminent breakthrough but voicing optimism that Qatar and its neighbours might soon be willing to talk face to face.
Sheikh Mohamed, who was in Ankara for talks with Turkish President Recep Tayyip Erdogan and Foreign Minister Mevlut Cavusoglu, said he believed the Kuwaiti-US mediation would "bear fruit in time".
Getting the parties in the Qatar dispute to talk directly to one another would be an important next step in resolving the crisis, the US State Department had earlier said after Tillerson ended three days of talks in Gulf countries. Reuters quoted the State Department as saying that there was subtle progress in efforts to resolve the Gulf crisis.
Turkey has called the four Arab states' blockade of Qatar unjust, and criticised their demands in order to end the sanctions, which include a requirement for Turkey's military base in Qatar to be closed.
Cavusoglu said the Turkish president hoped to visit the Gulf soon to discuss efforts to resolve the crisis. "All our efforts are focused on a solution that suits the laws of brotherly relations," he said.
Besides the latest developments pertaining to the GCC crisis, including all the illegal measures taken against Qatar, Sheikh Mohamed and Cavusoglu reviewed bilateral relations during their meeting yesterday and the means to enhance them, as well as a number of issues of joint interest, the official Qatar News Agency (QNA) reported.
Sheikh Mohamed expressed his appreciation of Turkey's position.
On his part, Cavusoglu stressed his country's position rejecting the siege on Qatar.
Turkish Prime Minister Benali Yildirim has said the measures taken by the siege countries against Qatar are neither fair nor correct. Qatar is a sovereign country, he stressed, and imposing dictations on it is unacceptable.
Yildirim said in a press statement yesterday that Turkey aims to end the tension in the Gulf region and prevent new crises among the countries of the region, according to QNA.
In response to a question on his intention or that of Erdogan to visit Qatar, Yildirim pointed to the possibility of holding meetings and conducting visits to the region in order to find a solution to the Gulf crisis.
Sheikh Mohamed briefed Turkish President Recep Tayyip Erdogan, during their meeting in Ankara yesterday, on latest developments in the Gulf crisis and the illegal measures taken against Qatar by the blockading countries.
The White House said US President Donald Trump spoke by phone to Saudi King Salman on Friday, according to Al Jazeera. The two leaders discussed efforts to resolve the month-long dispute between Qatar and the four Arab states.
HE the Foreign Minister Sheikh Mohamed bin Abdulrahman al-Thani (L) shaking hands with Turkey's President Recep Tayyip Erdogan at the presidential palace in Ankara.
HE the Foreign Minister Sheikh Mohamed bin Abdulrahman al-Thani has said it would be unfair to describe US Secretary of State Rex Tillerson's shuttle diplomacy to find a solution to the Gulf crisis as a failure, insisting that it "cannot be solved in a day".
