"Prior registration will make it easier for these people or their relatives to be contacted by SMS" in case of any emergency situations, a spokesman told AFP.
Currently some 44,000 tickets have been bought in Austria for matches in the June 10-July 10 championships in France, but many others are also expected to make the journey.
The foreign ministry's move comes after terror attacks in Paris on November 13, claimed by the Islamic State extremist group, including outside the Stade de France stadium during a football match.
A total of 130 people were killed in the attacks.
France is planning ultra-tight security for the competition, which is being played in venues nationwide involving 24 teams and several million fans.
French security services have carried out several large-scale exercises in recent weeks, including one in Saint-Etienne simulating a chemical attack.
The Austrian foreign ministry has set up a website, www.reiseregistrierung.at, where people can register online.
For data protection reasons, the data will be deleted 14 days after their return.