"It will be a routine inquiry... to know the actual reason behind the death of the soldier," an army spokesperson said. Sharma worked as a driver at a station in Rajasthan.
Tension prevailed in Patna following the encounter, which took place at Paliganj Bazaar on Tuesday.
The in-charge of the Paliganj police station, Krishna Deo Sinha, said Sharma was a criminal and an army-deserter. The police had received his arrest warrant from the army headquarters, he said adding that Sharma and two others were trying steal a banker's motorcycle.
He added that when they tried to stop Sharma, he opened fire on the police, who fired back. Sharma's family, however, denied the claims and said that the encounter was fabricated.
Sharma, who is survived by his wife and a two-year-old son, was cremated on Wednesday with military honours.
The killing came a month after the police shot down three students in Patna in another alleged staged gunfight.