It comes days after rebels said they were suspending their part in a peace accord signed with the government. Both sides claimed the other attacked first.
It was the first time the two sides have fought in the provincial capital since the announcement last week by ethnic Tuareg rebels.
Kidal has remained largely under the control of the separatists even after the return of the Malian military in June. Prior to that, the soldiers had retreated in the face of a 2012 rebellion by ethnic Tuaregs.
Resident Mohamed Cisse said: "We have heard rounds of automatic gunfire and have retreated inside the house.
"It's an exchange of fire between the NMLA who came to attack the Malian soldiers who were guarding the bank."
Several hours later Mr Cisse said that relative calm had returned.
The peace accord signed in June had allowed for the Malian army's return and for a July presidential election to proceed, the first since a March 2012 coup worsened the chaos in the country.
In the aftermath, Tuareg rebels and radical al Qaida-linked jihadists took control in northern Mali. Tuareg rebels later retreated until a French-led military intervention ousted the jihadists from the country's northern provincial capitals.