Temirlan M, 19, is in a coma after being shot in the head during a special forces operation in northern Georgia on December 26th, the Kronen Zeitung newspaper reported.
He is believed to be linked to Chechen warlord Akhmed Chatayev, who is suspected of organising a 2016 attack at Istanbul's airport.
Georgian authorities asked that the man receive emergency surgery in a Viennese hospital but a spokesman for Austria's interior ministry said the government had vetoed his entrance to the country.
The arrival of Temirlan M would pose a “major risk to domestic security,” the spokesman said, adding that possible rescue or blackmail attempts could be made if the man entered the country.
A country of 8.7 million people with a large Chechen community, Austria has one of Europe's highest numbers of foreign jihadists per capita.
Chatayev was killed during a counter-terror operation in Georgia's capital Tbilisi in November.
He had been identified by Turkish media as the organiser of a triple suicide bombing at Istanbul's main airport in June 2016 in which 46 people were killed and more than 200 wounded.
Austria's new government, a coalition between the conservative People's Party (ÖVP) and the far-right FPÖ entered office in mid-December, pledging to stop illegal immigration and to make national security a priority.
The FPÖ party runs the interior ministry.