But Qatar on Saturday rejected the demands as unrealistic, calling the blockade "illegal".
State Department Spokeswoman Heather Nauert said talks would continue through the week, but added the Saudi demands remained "challenging" for Qatar.
Tillerson added: "A productive next step would be for each of the countries to sit together and continue this conversation".
He added: "If you look at the words of Secretary Tillerson, you have to slightly read between the lines, because he has been very careful - all of the countries involved are all close USA allies". Tillerson also said a "lowering of rhetoric would also help ease the tension".
"The foundation of the dispute with Qatar is diplomatic and security-oriented, never military", Bahrain's foreign minister, Sheikh Khalid bin Ahmed al-Khalifa, said on Twitter.
Sever ties to all "terrorist, sectarian and ideological organizations", including the Muslim Brotherhood, Hezbollah, Islamic State, and Al Qaeda.
"Our demands on Qatar are non-negotiable".
The story raised concerns about the closure of Al Jazeera, citizenship rights and extraditions.
Four Arab countries demand from Qatar to comply with 13 points in return for an end to a three-week-old diplomatic and trade boycott of the country.
"In this context NHRC urged the state of Qatar not to accept those demands", the article said.
"To ask Turkey to pull out its troops from Qatar is firstly disrespectful behaviour towards us", he said in Istanbul, according to Reuters. "And when they react they're going to hurt Emiratis and in particular Saudi Arabia", Iqbal said.
US President Donald Trump has announced billions of dollars in arms sales to GCC members, and it was not immediately clear what deals could be impacted. Qatar said the website was hacked.
A handout picture provided by the office of Iranian President Hassan Rouhani shows him speaking at a conference in the capital Tehran.
A Corker aide said his action would not affect sales that had already been reviewed by Congress or non-lethal assistance, including training. Corker said Monday, June 26, 2017, that he'll withhold approval of USA weapons sales to several Middle Eastern allies until there is a clear path for settling a diplomatic crisis with Qatar.