The prime minister and opposition leader are pleased the Turkish government has seemingly toned down some incendiary comments made by the country's president.
Prime Minister Scott Morrison has welcomed the Turkish president toning down his language after an inflammatory speech that sparked a diplomatic row.
Mr Morrison believes Recep Tayyip Erdogan has "moderated" his views about Australians after a series of high-level crisis meetings overnight.
"It's my intention to break any cycle of recklessness and work through the issues practically," he told reporters in Melbourne on Thursday.
"To register in the strongest and clearest of terms the offence that was taken - I believe rightly, by those comments yesterday - but now to work constructively."
Turkish-Australian relations plummeted after Mr Erdogan threatened to send Australians with anti-Muslim views home in coffins, like their grandfathers.
Mr Erdogan was referencing the WWI battle at Gallipoli, in which thousands of Australian and New Zealand soldiers died fighting the Turks.
"Your grandparents came, some of them returned in coffins," the Turkish president said at a political rally, where he also played footage of the Christchurch mosque massacre.
"If you come as well like your grandfathers, be sure that you will be gone like your grandfathers."
The Turkish president's office has claimed his incendiary address was taken out of context.
"As he was giving the speech at the Canakkale (Gallipoli) commemoration, he framed his remarks in a historical context of attacks against Turkey, past and present," a senior aide tweeted.
Home Affairs Minister Peter Dutton said the comments were dangerous and very disturbing.
Mr Dutton said many Australians - including school groups - were in Turkey to visit war sites and memorials ahead of Anzac Day next month.
"If there is an apology and a back-down, I think that's entirely appropriate," he told 2GB radio.
Opposition Leader Bill Shorten is also pleased the Turkish government has adopted more "temperate, conciliatory" language.
"It's been made clear to me that Australians and New Zealanders are welcome, especially for the Anzac commemorations coming up," he told reporters in Perth.
The prime minister summoned the Turkish ambassador to Parliament House on Wednesday to explain the remarks.
He confirmed all options for responding to Turkey were on the table, including demanding a formal apology or kicking out the ambassador.
Australian diplomats are also attending an emergency meeting of the Organisation of Islamic Cooperation (OIC) in Turkey on Friday to discuss the terror attacks in New Zealand.
Meanwhile, official travel advice for Turkey is still under review, with thousands of Australians planning to go there for Anzac Day services.
Australia's travel advice for Turkey is already set at "exercise a high degree of caution" due to the threat of terrorism.
"I welcome some of the more positive statements about the safety of Australians travelling from the Turkish government overnight," the prime minister said.
"But I still think we have a bit more distance to travel."
Tour companies Fanatics and Intrepid, who take Australians to Gallipoli for Anzac Day, say their plans have not been rattled by the issue.
Australian white supremacist Brenton Tarrant, 28, has been charged with murder after 50 people were shot dead and dozens more injured at two Christchurch mosques last week.