Tillerson's comments came on May 20 in Riyadh during President Donald Trump's visit to Saudi Arabia, the first leg in the president's first foreign trip.
It also came a day after moderate Iranian President Hassan Rohani was reelected by a wide margin, defeating a more hard-line opponent - a victory that could reinvigorate efforts for an economic and diplomatic thaw with the West.
Tillerson said a centerpiece of Trump's visit to Saudi Arabia is to curb any threats to the region posed by Iran.
Sunni-led, U.S. ally Saudi Arabia and mainly Shi'ite Muslim Iran are bitter rivals for influence in the region.
Tillerson, when asked whether he would meet with Iran's Zarif, said he would "not shut out anyone who wants to talk" or have a productive conversation.
"In all likelihood," he added, "we will talk at the right time."
Speaking in a joint briefing, Tillerson and Saudi Foreign Minister Adel al-Jubeir criticized Iran for its support of foreign fighters in countries such as Yemen, Iraq, and Syria, and for its nuclear ambitions.
Tillerson called on Rohani to end Tehran's support and financing of terrorist groups, or what Tillerson called "destabilizing forces that exist in this region."
Comment: In reality, the US/NATO is the main destabilizing force in the Middle East, along with Israel and Saudi Arabia.
The United States and other critics in the West accuse Tehran of sponsoring international terrorism and destabilizing the region, and Iran is still targeted by U.S. sanctions over its weapons programs and perceived rights violations.
Trump was greeted warmly during his Saudi visit, and U.S. and Saudi officials signed at least $350 billion in business transactions on the first day -- including a $110 billion defense and military deal to bolster the Saudi military.
How, but how can he say that with a straight face ? And what does he call what the US are doing in Syria, just to name the last venture ? A humanitarian mission maybe ? I'm sure these people really think that we, what i call ordinary folks, cannot think for ourselves. I have to admit that I cannot think along the same lines as ''they''. Then again, maybe I would if I had more $$$ than i could spend. Who knows...