Armed police are at the scene in the centre of the city searching for the gunman and the Christmas market has been evacuated.
Security officials have cordoned off the area and trains have been stopped as they look for the gunman, who is known to the police .
The shooting has happened at a time when the European Parliament is in the city of Strasbourg for a meeting.
MEP Martina Anderson tweeted: “No confirmed reports but footage out of people on the ground injuries if not worse – My thoughts & prayers with all who are injured.”
The French Interior Ministry has urged people to stay inside and said there had been a “serious public security incident”.
Strasbourg deputy mayor Alain Fontanel tweeted: “Shooting in downtown Strasbourg. Thank you all for staying home while waiting for a clarification of the situation.”
The Strasbourg Christmas market is internationally popular and attracts millions of tourists annually.
Emmanuel Macron has been kept up to date and has sent his Interior Minister Christophe Castaner to the scene.
France continues to be on high alert after the country has suffered attacks in 2015 and 2016 killing more than 200 people.
An email sent to MEPs said: “There seems to be several gun attacks in parallel in the city centre of Strasbourg. In case you are in the city centre in a restaurant, please stay inside and don’t go out.
“The decision has been taken, as a precautionary measure, to close the European Parliament building in Strasbourg. We ask you to stay calm and stay safe wishing EP premises.”
The French Counter Terrorism Prosecutor is monitoring the situation in Strasbourg according to the Prosecutor’s Office.
The MEP Terry Reintke tweeted: “Strasbourg is a symbol of peace and unity. Nothing can ever change that.
“Love to everyone who is out in the city tonight. Stay inside and be safe.”
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.
“This is a serious incident and I advise all staff and visitors to stay at home or indoors for the moment.”
Jean-Claude Juncker, head of the European Commission, tweeted: “My thoughts are with the victims of the Strasbourg shooting, which I condemn with the utmost firmness.
“Strasbourg is an excellent symbol of peace and European Democracy. Values that we will always defend”.