In separate discussions, Trump spoke with Saudi Arabia's Crown Prince, Muhammad bin Salman, United Arab Emirates (U.A.E.) Crown Prince, Sheikh Muhammad bin Zayed al-Nahyan, and Qatar's Emir, Sheikh Tamim bin Hamad al-Thani.
Trump's call to Qatar's Sheikh Tamim bin Hamad Al Thani on Thursday came after he met the emir of Kuwait, whom he thanked for his vigorous yet thus far unsuccessful mediation. Alleging that the country supported extremist groups and was too close to Iran, the group had imposed economic sanctions on Qatar, worsening the diplomatic crisis in West Asia.
The Saudi Crown Prince is to discuss how to proceed with the Saudi-led bloc, according to the report.
The Qatari emir expressed in the Friday conversation his interest in discussing the demands of the four countries in order to achieve the best outcome for all states involved, Saudi-owned television Al Arabiya reported.
Also, on Saturday, SPA reported that Mr Trump called Saudi Arabia's Crown Prince to "discuss the friendly relations between the two countries and the latest regional and worldwide developments".
President Donald Trump on Thursday said he is prepared to personally intervene to resolve the months-long dispute between Qatar and several other Gulf neighbors if the countries do not reestablish diplomatic relations soon.
During the phone call, both the Saudi's crown prince and Emir of Qatar agreed on sending two Qatari envoys to engage in dialogue with the Arab countries, to pave the way for reconciliation and end the diplomatic boycott adopted in May.
"I would be willing to be the mediator", Trump said, adding he believed he could quickly resolve the dispute.
Instead, the four Arab states stressed that "the military option has not been and will not be considered" under any circumstances. Doha had rejected their 13-point list of demands and condemned them as an attempt to infringe Qatar's sovereignty.
Standing alongside Kuwaiti Emir Sabah al-Ahmad al-Jaber Al Sabah, the United States president then warned, "If they don't stop the funding of terrorism, I don't want them to come together".
"Qatar is not serious in dialogue and continues its previous policies".
Qatar hosts a huge American air base, home to the headquarters of Centcom - the regional command which leads operations against the Islamic State jihadist group.
Tillerson said the US had agreed to "expedite" delivery of F-18 Super Hornet aircraft, sold to Kuwait past year as part of a $10-billion arms deal, and both countries agreed to expand sharing of counter-terrorism and border security information.