The 46-year-old Hudson, who brought a special talent and style to Chelsea, Stoke City and Arsenal, was hit by a car while walking home in the East End of London on Monday night.
Hudson, who made his name in the Chelsea team which beat Real Madrid to win the European Cup-Winners' Cup in 1971, had earlier attended the Sports Writers' Association's annual awards at a hotel in Kensington.
He was said to be in a "critical" condition after undergoing a 14-hour operation at the Royal London Hospital for internal and head injuries.
His son, Alan Jnr, said that the family had been told to prepare for the worst, but now the signs were more hopeful.
"I had a phone-call at one this morning and have been at the hospital since," he said. "At five or six o'clock this morning we were told it didn't look like he was going to pull through.
"Me and my family sat in the chapel praying and he's turned it around somehow. They have stabilised the bleeding and got rid of a blood clot in the brain."
At one stage Hudson was in too bad a condition to be transferred from the operating theatre, where a specialist was called in to stabilise his condition.
His wife, Ann, who married her husband in Bermuda six months ago, said she did not know exactly how the accident happened.
"According to the police, Alan had apparently stepped off the pavement and was hit by a car. I can't believe it. We were only just getting ready for Christmas together - and now this."
Hudson's former wife, Maureen, said she has been warned that her ex-husband, whose injuries include a shattered pelvis, has a 50-50 chance of survival.
"It's awful. When I first rang the surgeon, he said it was touch and go, but now he says he has a 50-50 chance of pulling through. Alan's a fighter," she said.
Glenn Hoddle, the England coach, said: "I am deeply shocked by the news and can hardly believe what has happened. Alan was one of the outstanding players of his generation."
Graham Kelly, the chief executive of the Football Association, added: "This is terrible news and everybody at the FA is stunned. We can only hope and pray that Alan pulls through."Reuse content