Ninety-eight-year-old Singh died this evening at a hospital here after suffering a cardiac arrest.
"Sad at demise of a great air warrior & Marshal of the Air Force Arjan Singh. Condolences to his family & IAF community.
"Marshal of the IAF Arjan Singh was a WW II hero & won our nation's gratitude for his military leadership in 1965 war," Kovind said in a series of tweets.
The prime minister said the determined focus of Singh, who was promoted to the five-star rank of Marshal of the Air Force in 2002, on capacity building in the IAF added great strength to India's defence capabilities.
"India will never forget the excellent leadership of Arjan Singh in 1965, when the IAF saw substantial action," he said.
Modi also recalled how the ageing war hero remained a consummate soldier till the very end.
"Sometime back I met him, who despite his ill health tried to get up to salute even though I said no. Such was his soldier discipline," he said.
Modi said his thoughts were with Singh's family and those mourning the demise of the distinguished air warrior and a fine human being.
Besides them, Defence Minister Nirmala Sitharaman, Home Minister Rajnath Singh, Finance Minister Arun Jaitley, BJP chief Amit Shah and chief Congress spokesperson Randeep Surjewala were among those who condoled the death of the war veteran.
"It is very rare for any country to have a soldier of that kind of credentials. It is a great loss for the country," Sitharaman said.
She too remembered how the war hero would always comport himself like a true soldier.
"I have seen him standing up and saluting the president and prime minister in official functions. He would never forget the decorum of a much-acclaimed soldier who knew what it takes to be in the armed forces," the defence minister said.
Rajnath said he was deeply pained at Singh's demise.
"Marshal of the IAF, Arjan Singh lived his life like a soldier. The tales of his courage will remain immortal. Deeply pained at his demise.
"My heartfelt condolences to the bereaved family members of Shri Arjan Singh. May God give them strength to bear this colossal loss," the home minister said.
Jaitley said Singh was one of the country's greatest soldiers in recent memory.
"Condolences on the sad demise of Marshal of the Indian Air Force Arjan Singh ji. He was one of the greatest soldiers in the recent memory (sic).
"My thoughts & prayers are with the family of Marshal of the IAF Arjan Singh ji. A grateful nation will miss him," the finance minister said on Twitter.
Shah, meanwhile, hailed Singh's rich contribution to the motherland.
"Pained by the sad demise of Marshal of the Indian Air Force Arjan Singh ji. I express my deepest condolences to his family and friends.
"Marshal of IAF Arjan Singh ji led the Indian Air Force through his exemplary leadership. His rich contribution to motherland is venerable," Shah said in a series of tweets.
Surjewala said the country has lost a great soldier and a fine diplomat.
"Deeply saddened by the passing away of @_IndianAirForce Marshall #ArjanSingh ji. India has lost a great soldier and a fine diplomat today," he tweeted.
Goa Chief Minister and former defence minister Manohar Parrikar said the nation has lost a great icon.
"As defence minister, I saw him on a number of occasions and always wondered how a man of his age could think of modernisation of the air force and talk about defence, the air force in particular. We have lost a great icon," Parrikar said.