While part of South Ossetia's former Soviet borders, Akhalgori was under Tbilisi's control after the region broke away from Georgia in the early 1990s.
Interfax quoted Alexander Drobyshevsky, the Russian defence ministry spokesman, as saying Georgian officials may have pressured Glukhov after his abduction into making statements discrediting the Russian military.
The Russian news agency said Drobyshevsky had demanded that Georgia hand the soldier over immediately.
Earlier Glukhov was shown on Georgian television, dressed in military uniform.
He said he was asking Georgia for asylum, saying conditions were unbearable in the Russian army.
"I am asking the president of Georgia to grant me asylum. I cannot bear the hardship Russian soldiers are experiencing here anymore. I hope asylum will be granted," he said.
Glukhov said he was first based in Tskhinvali, the South Ossetian capital, and then transferred to Akhalgori.
Shota Utiashvili, a Georgian interior ministry official, said Glukhov had handed himself over to Georgian police on Monday, complaining that the major of his unit had been beating him.
Utiashvili said: "We did not detain him. He is free and can do whatever he wants. He can go back to Russia, or stay in Georgia."