Latest: The death toll has risen to four according to local media quoting police sources.
It was also reported that the Strasbourg home of the 29-year-old suspect was searched by police executing a warrant relating to a robbery on Tuesday morning but he was not there.
Grenades were found by police, BFM TV said.
Update 10:24pm: A suspected extremist is on the run after a shooting which left two people dead near a Christmas market in the French city of Strasbourg.
Live from Strasbourg pic.twitter.com/MF28PUaMc3— 👊 𝚁𝚘𝚋𝚋𝚎𝚛𝚝 𝙱𝚊𝚛𝚞𝚌𝚑 👊 (@RBaruch) December 11, 2018
French authorities have launched a terror investigation into the shooting which left seven people in a serious condition and four others injured.
The office of the Paris prosecutor said it was investigating a terrorist murder and attempted murder.
The gunman has been identified, has a criminal record and, according to the prefect of the Strasbourg region, had been flagged as a suspected extremist.
It is reported he was injured after exchange of gunfire with a soldier.
The suspect was said to have entered central Strasbourg via the Corbeau bridge to the south of the city centre, before heading to Rue des Orfevres, a popular shopping street close to the cathedral, where he opened fire.
There were unconfirmed reports of people being injured in several streets south of Place Kleber in central Strasbourg.
It is thought he may then have headed south of the city centre to the Neudorf or Place de l'Etoile area where people were advised to stay indoors.
The European Parliment, which is around two miles away from the city centre, was in lockdown and French President Emmanuel Macron adjourned a meeting at the presidential palace to be able to monitor the events.
In a statement the prefect of France's Bas-Rhin region confirmed that two people had died, seven were in a serious condition and four others had been injured in the Strasbourg shooting.
The prefect of France's Bas-Rhin region said the gunman, who is still at large, has been identified.
Authorities have not given a motive for the shooting.
The European Parliament spokesman, meanwhile, said that the building was on lockdown in Strasbourg.
Jaume Duch said "the European Parliament has been closed and no one can leave until further notice".
In a statement Sinn Féin MEP Martina Anderson said: "Tonight's shooting incident at the Christmas markets at Place Kleber in Strasbourg has shocked the city.
"I was in the city centre at the time and heard the gunfire and people, including young children, running away in panic.
"This incident has caused panic in the area with crowds of people out enjoying the Christmas markets.
"Details of the incident are still emerging but my thoughts are with those injured and all of those caught up in this incident."
Reports from France say shots have been fired in the north-eastern city of Strasbourg.
It is believed to have happened close to a popular Christmas market.
The Reuters news agency is reporting one person has been killed and six others are injured, but this has not been confirmed.
Police in Germany have said they are strengthening controls at the Franco-German border near Strasbourg.
The police force of Baden-Wurttemberg, a state in south-west Germany bordering Strasbourg, tweeted they were taking the extra measures at the border because of the shooting.
UPDATE: One dead, six injured in gunshot attack in Strasbourg, gunman on the run - French police sources pic.twitter.com/TKKjsKoet7— Reuters Top News (@Reuters) December 11, 2018
The French Interior Ministry is urging people there to stay indoors.
Sinn Féin politician Martina Anderson, a Member of the European Parliament representing Northern Ireland, tweeted: "We were in the centre of Strasbourg town when gun shots went off."
She added: "My thoughts & prayers with all who are injured."
We were in the centre of #Strasbourg town when gun shots went off
No confirmed reports but footage out of people on the ground injuries if not worse -
My thoughts & prayers with all who are injured.#StrasbourgMarket— MEP Martina Anderson (@M_AndersonSF) December 11, 2018
The European Parliament building is around two miles from the centre of Strasbourg.
Local authorities in the Grand-Est and Bas-Rhin region tweeted for the public to "avoid the area of the police station", which is close to the city's Christmas market.
Thorbjorn Jagland, Secretary General of the Council of Europe, based in Strasbourg, wrote on Twitter: "The police informed me tonight of a shooting in Strasbourg with possibly one victim and several wounded.
"The perpetrator is still on the run.
The Police informed me tonight of a shooting in Strasbourg with possibly one victim and several wounded. The perpetrator is still on the run. This is a serious incident and I advise all staff and visitors to stay at home or indoors for the moment.— Thorbjørn Jagland (@TJagland) December 11, 2018
"This is a serious incident and I advise all staff and visitors to stay at home or indoors for the moment."
Several other MEPs tweeted to say that both they and their staff were safe and accounted for.
Strasbourg Christmas market is one of the oldest in Europe with 300 wooden chalets set up in the city’s historic centre from November 23 to Christmas Eve.
One of the biggest Christmas trees in Europe is put up in Place Kleber, the largest square in the city, which was named after French general Jean-Baptiste Kleber who was born in Strasbourg in 1753.