The Tibesti mountains near the Libyan frontier frequently see fighting between rebels and the army, as well as in the northeast bordering Sudan. French air strikes were needed to stop an incursion there in February 2019.
On April 11, the Union of Resistance Forces (UFR), led by Deby's nephew Timan Erdimi, entered Chad from Libya in columns of pick-up trucks.
The UFR said Sunday that it supported FACT, but without saying whether its fighters took part in Saturday's clash.
The UFR was allied with Chadian rebels in 2008 when they launched a coup attempt, stopped at the gates of the presidential palace by a French military intervention.
Deby assured re-election
FACT is mainly made up of the Saharan Goran people, while the UFR is majority Zaghawa, the president's ethnic group.
The US embassy in N'Djamena on Saturday ordered non-essential personnel to leave the country, warning of possible violence in the capital.
Britain also urged its nationals to leave.
But Chadian Foreign Minister Amine Abba Sidick said Sunday: "The military situation is well under the control of the Chadian army."
Meanwhile vote-counting was continuing from the April 11 election in which Deby is assured victory after sidelining his main rivals.
Deby, who has ruled the former French colony with an iron fist and is a key Western ally in the fight against jihadists in western and central Africa, campaigned on security issues in the troubled region.
Preliminary results will not be announced until April 25.