Turkey has been carrying out a series of retaliatory strikes against President Bashar al-Assad's forces fighting rebels along the border since Syrian shelling killed five Turkish civilians in a Turkish frontier town at the start of October.
Tension between the two neighbours, once close allies, is at its highest since Ankara turned against Assad last year over his violent crackdown on anti-government protests.
The report in the daily Milliyet newspaper, written by columnist Fikret Bila who is known to have good contacts with the military, cited unnamed military sources. Turkey's military, which rarely talks in public, could not be reached for comment.
The report said the retaliatory fire had been in response to 27 mortars or shells fired from Syria. Turkey had responded to every incident, it said.
Twelve Syrian soldiers had been killed as a result of Turkish fire, the report said. Five Syrian tanks, three armoured vehicles, one mortar weapon, one ammunition vehicle and two anti-aircraft guns had also been destroyed and many other military vehicles had been damaged.
Eighteen mortar shells fired from Syria had landed in the Akcakale district of Sanliurfa province, where five Turkish civilians were killed this month, while nine had landed in Hatay province further to the west, according to the report.
Turkey had fired 69 times from Hatay and 18 times from Akcakale, it said.
The report also stated that Turkish F-16 war planes were on high alert at the Incirlik air base in Adana, some 100 km (62 miles) from the Syrian border. The fighters had been scrambled as recently as Friday in response to Syrian helicopters flying close to the shared border, it said.
Turkey's Chief-of-Staff General Necdet Ozel said this month that his troops would respond "with greater force" if shells continued to land on Turkish soil, and parliament has also authorised the deployment of troops beyond Turkey, heightening fears that Syria's civil war could drag in regional powers.
Turkish Foreign Minister Ahmet Davutoğlu called on Friday for all sides involved in the Syrian conflict to observe a ceasefire during the Islamic Eid al-Adha festival next week.