“There are several possibilities on the table… Electronic ankle tags for people posing a potential threat are definitely something to be considered,” Hans Peter Doskozil told Austrian radio.
“I definitely think that it would be appropriate that… they are monitored, that the authorities know what they are doing, whom they are meeting. It's very important,” he said.
Austria's centrist government has been preparing a raft of measures to beef up security and reduce illegal immigration, including tighter border controls and more video surveillance.
Interior Minister Wolfgang Sobotka told reporters before a cabinet meeting on Tuesday that the use of electronic tags, which he also supports, would require new legislation.
Around 300 people have either left or were intercepted trying to leave Austria to fight in Syria, according to the interior ministry. Around 40 have died there while some 90 have come back.
On Friday, Austrian police detained a 17-year-old they suspected of being an Islamic extremist in Vienna. The authorities believe he may have been planning a bomb attack in the capital.
This was followed on Saturday by the arrest of a 21-year-old man in the western German city of Neuss.
Media reports said that he and the Austrian, identified in reports as Lorenz K., had experimented together with making explosives in Germany.
Sobotka said on Monday that Lorenz K., an Austrian citizen who was born and raised in Austria, had told investigators that he supported the extremist Islamic State group.
On Tuesday, he said the suspect admitted to having built an experimental bomb. But no bomb-making equipment was found at his flat, the Austria Press Agency reported.
Austrian police have raided other addresses, and on Monday public security chief Konrad Kogler said that a minor — reportedly a 12-year-old boy — had been questioned in connection with the enquiry.
Austria has so far been spared the string of attacks by Islamist extremists in other European countries in recent years.