The official-level meetings aim to restore relations Saudi Arabia severed five years ago after protesters in Iran stormed its diplomatic missions following the execution of a revered Shia cleric in the Arab kingdom.
"We have initiated some exploratory talks. They are at a very early stage but we are hopeful," Prince Faisal bin Farhan told AFP in Paris on Tuesday, where he was participating in two international summits.
The talks with Iran, facilitated by Iraqi Prime Minister Mustafa al-Kadhimi, had remained secret until the Financial Times reported that a first meeting had been held in Baghdad on April 9.
The Iranian government confirmed on May 10 that the talks had been held, adding that it was "too soon" to discuss the results.
"The purpose of the talks was both bilateral and regional," Foreign Ministry spokesman Saeed Khatibzadeh told reporters.
"But let us wait and see the results of these talks ... it might still be too soon to talk about the details of the negotiations," Khatibzadeh added, noting that Iran has "always welcomed such talks at any level and in any shape".
Khatibzadeh noted that “de-escalation and establishing ties between two great Islamic countries in the Persian Gulf region is to the benefit of both nations".
Asked what the impact of the June presidential elections in Iran might be, the Saudi diplomat said he thought it would be minimal.
"Our understanding of Iran's foreign policy is that it's set by the Leader," Prince Faisal explained. "So we don't think there will be a substantial change."
"There may be a change in the representatives that deliver the policy, but in the end, it's what happens on the ground that matters."
Iraq's President Barham Salih said later that Baghdad hosted more than one round of talks between Saudi Arabia and Iran.
Rayed Krimly, the head of policy planning at the Saudi Foreign Ministry, said earlier this month that the talks are aimed at reducing regional tensions.
"As to current Saudi-Iranian talks they aim to explore ways to reduce tensions in the region," Krimly said.
"We hope they prove successful, but it is too early, and premature, to reach any definitive conclusions. Our evaluation will be based on verifiable deeds, and not proclamations."
In April, Saudi Arabia's crown prince struck a conciliatory tone, saying he sought "good" relations.
The initiative comes at a time of shifting power dynamics.
US President Joe Biden is seeking to rejoin the tattered 2015 Iran nuclear deal that was abandoned by his predecessor Donald Trump.
Meanwhile, Washington-Riyadh relations have cooled from the very close ties of the Trump era, as Biden seeks to press the oil-rich kingdom over human rights concerns.