The foiled robbery took place at a BNP bank branch, 400 metres (1,300 feet) from the Arc de Triomphe monument, one of the citys biggest tourist attractions which commemorates the countrys military victories and war dead.
A 24-year-old man burst into the bank on Avenue de la Grande Armee shortly after it opened. The alarm was raised and the police were waiting for the man when he came out brandishing a gun.
The officers fired at the man, injuring him in the leg and chest. He was taken to hospital.
Paris has been the scene of several spectacular robberies in recent years, mostly involving jewels.
US reality TV star Kim Kardashian was famously targeted in a diamond heist in the French capital in October 2016.
She was tied up and held at gunpoint in a luxury apartment while robbers made off with jewellery worth at least nine million euros ($11 million).