A Turkish Foreign Affairs source has characterised the message from New Zealand officials as: "We will come and take her. Give us some time because of the Covid-19 threat," according to RNZ correspondent Vehbi Bas.
Aden acquired dual nationality after moving to Australia when she was 6 and travelled to Syria from there in 2014, on an Australian passport, to join Islamic State.
That led to Australia stripping her of citizenship last year, enraging New Zealand.
The Ministry of Foreign Affairs (MFAT) said: "Nothing has yet been agreed or determined.
"As the woman involved in this case is a New Zealand citizen, we continue to engage with Turkish authorities on practical matters."
Those practical matters could include travelling here from Turkey; it would most likely be a commercial flight and she'd have to be accompanied by New Zealand law enforcement officials.
Arrangements would also have to be made for two weeks' isolation at the border, secure enough to convince the public she posed no risk.
Her case sparked a diplomatic row with Prime Minister Jacinda Ardern accusing Australia of "abdicating" its responsibility.
There's no set timeframe for Aden's possible return to New Zealand, but there is an expectation the situation should be resolved either way within the next month or so.
About three weeks ago RNZ reported Turkey had dropped charges against her and had begun procedures for deportation.
Ardern has made it clear the involvement of two children would heavily influence New Zealand's response, but national security would still be paramount.
In its statement, MFAT also said, as had "previously [been] made clear, New Zealand is in discussion with Australia about how this case should be managed".
"This includes the issue of where the woman and children will end up."
Ardern spoke to Morrison the evening she publicly lashed him and said recently officials on both sides of the ditch were working on a possible solution after the leaders' joint statement: "Regardless of the steps taken in this case to date, both NZ and Australia acknowledge that this case now has a number of complexities", pledging to work together in the "spirit of the relationship".
The reality is unless there's a sudden change of heart from Australia, Aden and her children are New Zealand's responsibility through her sole citizenship and that's where they'll end up.
Terrorism laws in place
Correspondent Bas also cites a source at Istanbul Police Headquarters saying Aden had not been questioned by Turkish terrorism police, which meant any investigations or trials would be held wherever she ended up.
There's been a wave of anti-terrorism laws passed in New Zealand since the early 2000s; first in reaction to the 911 attacks, and then as ISIS took hold in the Middle East and waged attacks across the globe. The most recent was a law giving the Government the ability to track and manage those returning from foreign conflicts, suspected of terrorist activities.
The Terrorism Suppression (Control Orders) Act 2019 was passed in a rush when New Zealander Mark Taylor surfaced in Syria.
It gives authorities the ability to pre-emptively track and monitor people, but under strict thresholds, including that they pose a "real risk of engaging in terrorism-related activities".
When deciding whether an order should be imposed, a High Court judge has to deem it necessary to "protect the public from terrorism" and to "support the relevant person's reintegration into New Zealand".
While there's a long list of restrictions: being at certain places, not leaving the country, not associating with certain people, alongside requirements like checking in with police and being fingerprinted, there is no specific reference to detaining someone on arrival.
Under the Terrorism Suppression Act 2002, you can be jailed for up to 14 years for knowingly "participating in terrorist group", even if it happened overseas; but that relies on the offences being proved through the New Zealand justice system.
