Singhania, 67, hit 21,000m (69,000 feet) a little more than two hours after taking off in his 40-ton balloon on Saturday, and has started his descent, his son Gautam said.
"We have the world record at 69,000 feet," said Gautam Singhania. "We're bringing him down now."
The previous world record was 19,811m (64,997 feet), set by Sweden's Per Lindstrand in Plano, Texas, in June 1988.
Singhania lifted off from downtown Mumbai (formerly Bombay) at 6.45am (0115 GMT) enclosed in a pressurised cabin suspended from the 50 metre (160 foot) multi-coloured balloon.
He had hoped to reach 21,336m (70,000 feet), but one of the balloon's designers said he could not quite make it.
"He must have hit a pocket of cold air, so he decided to quit while ahead," said Colin Prescott.
"The exact height reached was 21,290.89m (69,852 feet). This is subject to certification," Prescott added.
The record could not be independently verified immediately.
A sealed altimeter, a GPS (global positioning system) unit and a barograph were installed inside the capsule to determine the balloon's height, Prescott said.
On landing, the seals will be broken and the instruments submitted to the Guinness Book of World Records for verification.
Singhania was expected to take two hours to get back to Earth, probably landing in Nashik, a town about 100km northeast of Mumbai, depending on wind speed and direction.
The record will be verified once
the balloon lands back on Earth
Before taking off in downtown Mumbai cheered by hundreds of residents and a marching band, Singhania said he expected to break the record.
"I'm very excited but very nervous and confident I'll make it," Singhania said.
The entire voyage was broadcast live on national television using cameras on board the helicopter, inside the balloon's cabin and on the ground.
The temperature outside the balloon was expected to fall as low as -93C (-135F) and oxygen was negligible.
Before taking off, Singhania stressed the importance of the pressurised cabin, saying that if a person were exposed to such temperatures his "blood would boil."
A unit on the ground was monitoring oxygen levels and other life support systems inside the cabin.
The cabin also has a parachute, so if Singhania loses consciousness or any other high-altitude emergencies occur, the parachute will be automatically released and the cabin will drift to the ground, the second designer Andy Elson said.
Singhania, the chairman emeritus of the Raymond Group, one of India's leading textile companies, also set a record for ultra-light aviation 17 years ago when he flew 9,655km from Britain to India in 23 days.