King and Firkin has been held as prisoners for about 10 weeks after being arrested for reportedly flying a drone without a permit, the Australian Broadcasting Corporation (ABC) reported.
They have been travelling around the world since 2017 and documenting their adventures online.
The Australian Department of Foreign Affairs on Thursdau released a statement from the couple's families.
"(We) hope to see Mark and Jolie safely home as soon as possible," the statement quoted their families as saying.
Australian Prime Minister Scott Morrison said in a press conference on Thursday that government officials were making the necessary diplomatic contacts and following the case closely.
King is a dual Australian-British citizen and the UK Government has demanded the couple's release.
The British Foreign Secretary Dominic Raab called in the Iranian Ambassador on Wednesday.
The UK Foreign Office said Raab "raised serious concerns about the number of dual national citizens detained by Iran and their conditions of detention".
British Defence Secretary Ben Wallace said: "The message to Iran is to follow the international rule of law... and to release people that have been detained."
Another Australian-British dual citizen, a Melbourne-based academic, is also being held by Iran and has reportedly been sentenced to 10 years in jail, the ABC reported.
All three Austral are held at the notorious Evin jail, north of Tehran, which is used to house political prisoners.
Several dual nationals have been detained in Iran in recent years, including the British-Iranian woman Nazanin Zaghari-Ratcliffe, whose husband, Richard Ratcliffe, told the ABC that conditions inside the Evin facility were horrific.
( With inputs from IANS )