The woman was kidnapped yesterday near Koutiala, a city some 400 kilometres east of the capital, Bamako, which is in the Sikasso region near the Burkina Faso border.
"The Malian army is in pursuit," a security source who asked not to be identified told AFP.
A local official said the kidnappers drove away in a vehicle owned by the nun's Franciscan order.
There was no immediate claim of responsibility for the kidnapping which follows the Christmas Eve capture of a French aid worker, Sophie Petronin, in the restive north of Mali, where jihadists have staged repeated attacks.
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The Al-Qaeda in the Islamic Maghreb (AQIM) video showed Stockly, who was captured in the north, saying she was in good health.
The north fell under the control of Tuareg-led rebels and jihadist groups linked to Al-Qaeda in 2012 who were largely ousted by a French-led military operation in January 2013.
But the implementation of a peace accord struck in 2015 has been piecemeal with insurgents still active across large parts of the region.