Apart from the months-long crisis, Iran's rising influence in the Middle East is also expected to figure high on the agenda of America's top diplomat during talks in the two capitals.
Doha denies the charges and has rejected their terms for a settlement.
Tillerson made an unsuccessful attempt to resolve the dispute during a trip to the region in July.
US President Donald Trump, after initially appearing to support the effort to isolate Qatar, has called for mediation and recently predicted a rapid end to the crisis.
"I do not have a lot of expectations for it being resolved anytime soon," he said in an interview with financial news agency Bloomberg.
"There seems to be a real unwillingness on the part of some of the parties to want to engage."
Aside from the Gulf dispute and Iran, the conflict in Yemen and counter-terrorism will also figure in his talks, the State Department said.
While in Riyadh, Tillerson will also take part in the first meeting of a Saudi-Iraqi coordination council.
Iraqi Prime Minister Haider al-Abadi is leading a top-level ministerial delegation at the meeting, in a sign of warming ties as Sunni-ruled Saudi Arabia seeks to counter Tehran's influence in Shiite-majority Iraq.
On the Gulf crisis, the goal will be to try to persuade the two sides to at least open a dialogue.
After holding a working dinner with his Saudi counterpart Adel al-Jubeir on Saturday night, Tillerson was scheduled to meet other Saudi leaders Sunday before heading for Doha.
During his trip Tillerson is also to visit New Delhi in order to build what he said in a recent speech could be a 100-year "strategic partnership" with India.
Tillerson will stop in Islamabad to try to sooth Pakistani fears about this Indian outreach, but also pressure the government to crack down harder on Islamist militant groups.