Malian troops were fighting Islamists in Gao on Thursday.
The clashes left the main courthouse in flames.
Troops from Niger were the first to respond after heavy weapons fire was heard at at least two entries to the town overnight.
By midday Thursday the Malian army was battling what one military source said was a group of around 40 insurgents.
The source said at least three rebels had already been killed in the fighting.
One person was killed and another injured by the Kidal bomb, according to a journalist in the town speaking to the Reuters news agency.
Other witnesses told the AFP agency that two civilians had been injured.
The blast was within a kilometre of the camp where French and Malian soldiers have been stationed since they arrived in the town on 29 January, following its takeover by dissident Islamists and Tuareg separatists from Al Qaida-linked armed groups.
A French soldier was killed on Tuesday in fighting in the Ifoghas mountains, north of Kidal, where the hardline Islamists have taken refuge. About 20 rebels were also killed, according to the French military.