Insurgents led by al-Qaeda offshoot Islamic State of Iraq and the Levant (ISIS) seized 49 consulate staff and 31 other Turks on Wednesday.
The government in Ankara says it is watching the situation closely and the safety of the hostages is a priority.
But it has held back from calling for help from its Nato allies.
Among the 49 staff seized from Turkey's diplomatic buildings in Mosul are the consul general, Ozturk Yilmaz, members of Turkish special forces and some children, reports say. They are believed to be at a militant base in the city.
An Iraqi employee of the consulate told Turkey's Hurriyet daily on Thursday that "they are in good health - they are expecting to be released soon". He managed to contact them after they were taken hostage.
The other Turkish nationals are said to be lorry drivers who were taken hostage at a power plant in Mosul, where they had gone to deliver diesel.
Turkey has threatened to retaliate if any of its citizens are harmed and its foreign ministry says "all resources have been mobilised" for their safe return.
Emergency government meetings have been held and Nato ambassadors were summoned in Brussels, but Turkey has stopped short of invoking the founding treaty which member states can use if they feel their security is threatened.
After the meetings in Ankara, involving Prime Minister Recep Tayyip Erdogan, security chiefs and President Abdullah Gul, officials said they had agreed to try diplomatic methods to bring about the hostages' release.
However, ministers are said to be assessing military options too.
According to Justice Minister Bekir Bozdag, officials are looking at the terms of an earlier parliamentary mandate that sanctioned cross-border operations in Iraq.
The mandate expires later this year. and Mr Bozdag told reporters on Thursday that Ankara was not working on a new mandate, Reuters news agency reports.
The Turkish government says it will do whatever is necessary to secure the release of 80 Turks abducted by Islamist militants in northern Iraq.