About 720 security forces were deployed to search for Chekatt, French Interior Minister Christophe Castaner told the National Assembly in France on Wednesday.
The shooting is being treated as a terrorist attack, the Paris counter-terrorism prosecutor's office told ABC News.
Police said Chekatt fired shots in three separate locations near the center of the city on the Rue des Orfèvres around 8 p.m. local time.
Members of the French military confronted the suspect between 8:20 p.m. and 9 p.m. and used their weapons twice in an attempt to intercept him, Castaner said.
Chekatt fled the scene by jumping into a taxi, telling the driver that he had shot a soldier and injured 10 people. He also said that police had searched his home earlier and found a grenade and four knives, according to Heitz.
French soldiers shot him, but he fled the scene, officials said. Witnesses said the suspect used a handgun, but authorities haven't confirmed those reports.
On Tuesday morning, hours before the attack, police conducted a raid in which they intended to arrest Chekatt in relation to an attempted homicide investigation. Five other individuals were arrested during the operation, but the suspect was not found, Laurent Nunez, Deputy Interior Minister of France, said in an interview with radio outlet France Inter Wednesday morning.
Local schools are open on Wednesday, but parents also have the option to keep their children at home, Castaner said, adding that extra security will be patrolling streets and Christmas markets.
A photo posted to Twitter shows people locked down inside the European Parliament building in Strasbourg after the attack Tuesday.
French President Emmanuel Macron encouraged citizens to show their solidarity in the aftermath of the attack Wednesday.
ABC News' Ben Gittleson and Joshua Hoyos contributed to this report.