According to the (IFF), the two parties agreed in a meeting on Friday to finalise the 61-year-old's new deal on 1 August.
“We have to negotiate about the contract in the details. I wish to stay as Iran coach," Queiroz was quoted as telling Fars, Iran's news agency.
"The Iranians will always stay in my heart. I like to continue my job but the deal has not yet finalised.”
The former Manchester United assistant has been strongly linked with the vacant Bafana post, a position he held between 2000 and 2002.
His contract with Iran ended after the 2014 FIFA World Cup in Brazil.
Iran finished last in group F.
On June 20, in Brazil, Queiroz told the media he was stepping down as Iran's number one.
As Bafana coach, Queiroz successfully guided South Africa to qualify for the 2002 Africa Cup of Nations and 2002 FIFA World Cup, but resigned from the post before the start of the global spectacle, following a public fallout with SA Football Association (SAFA) hierarchy.
SAFA are due to announce Gordon Igesund's replacement as Bafana head coach next Saturday, five days before Queiroz's proposed new deal.