Argentine President Mauricio Macri made the announcement to a group of relatives of ARA San Juan's 44 crew members at his official residence on Tuesday.
Defence Minister Oscar Aguad later said the reward could be around $4m, but the exact figure will be announced later this week.
Family members pushed for Mr Macri to extend the search area and for private companies to be allowed to look for the naval vessel.
The TR-1700 electric diesel submarine disappeared on 15 November while sailing "on a routine training exercise" off te eastern coast of Argentina.
Relatives and friends of a crew member who missing
So far, 13 countries have been involved in the hunt, but to no avail.
There was hope on Christmas Eve when a sonar search by the remotely operated Russian Panther Plus submarine detected a new contact in the South Atlantic.
However, nothing came of it.
In late November, families were told the navy had abandoned hope of finding the crew alive after the submarine captain's last message reported seawater had entered the ventilation system, causing a battery to short-circuit and start a fire.
A major search effort involving 13 countries was launched
Officials also said an unusual sound was detected in the ocean near the submarine's last known position - three hours after it made its final contact.
They said the noise was "consistent with an explosion".
Previous detected contacts have been false alarms, with one coming from a sunken fishing boat.