The situation comes amid growing tensions between the West and Iran.
Several dual nationals have been detained in Iran in recent years, including the British-Iranian woman Nazanin Zaghari-Ratcliffe.
Relations between the UK and Iran have also been strained in recent months by a row over the seizure of oil tankers in the Gulf.
Australia also announced in July that it would join the US and the UK in policing the Strait of Hormuz against Iranian threats.
Ms King – who has dual UK and Australian nationality – and Mr Firkin, an Australian citizen, were reported to have been travelling on Australian passports.
In 2017, the couple left Western Australia to embark on a major trip driving across Asia to the UK. They were documenting their adventures on Instagram and Youtube, where they had more than 20,000 followers.
Videos of their travel through a dozen countries featured their cultural interactions and often showed drone footage of the natural landscape.
“Our biggest motivation… is to hopefully inspire anyone wanting to travel, and also try to break the stigma around travelling to countries which get a bad wrap [sic] in the media,” the pair wrote online.