Modi went to Army's Research and Referral Hospital here to see Hanumanthappa soon after he was airlifted from Siachen and prayed for his best.
"No words are enough to describe the endurance & indomitable spirit of Lance Naik Hanumanthappa. He is an outstanding soldier," he tweeted.
"Team of doctors is attending to Lance Naik Hanumanthappa. We are all hoping & praying for the best," Modi added.
His condition is critical but stable and he is undergoing various tests in the hospital, army sources said.
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Just before visiting the hospital, the Prime Minister tweeted: "Going to see Lance Naik Hanumanthappa, with prayers from the entire nation."
Hanumanthappa, who hails from Karnataka, was found alive yesterday, buried under 25 feet of snow for six days after an avalanche hit his post at the altitude of 19,600 feet close to the Line of Control (LoC) with Pakistan. The temperature at that altitude was minus 45 degrees Celsius.