The former Manchester United and Aberdeen manager is said to have spoken to his family and doctors but remains seriously ill in hospital after the operation following a fall on Saturday.
It was previously confirmed by a spokesman that the procedure had gone well, but the 76-year-old still needed “a period of intensive care to optimise his recovery”.
However, the latest reports suggest the Glaswegian awoke from a coma on Monday and has been asking medical staff about his condition.
Fife provost and councillor Jim Leishman, director of Dunfermline Athletic Football Club, is an old friend and football associate.
The pair spent time together just a few days ago, when Sir Alex – who played for Dunfermline in the 1960s – was guest of honour at 50th anniversary celebrations for the club’s Scottish Cup win on 28 April. Mr Leishman, who at 28 became the youngest manager in Scottish football when he took over at Dunfermline in 1983, said he hadn’t yet spoken to Sir Alex but had been told by mutual friends that his condition had improved.
“His brother has spoken to one of my friends and I heard that he was talking. I think he may even have got up, but he’s definitely talking.
“I’ve not spoken to him yet – when you have someone like Alex Ferguson, the whole world wants to talk to him.
“I certainly don’t want to pester him. I’ll just give him some time and wait until he recovers and gets a bit stronger before getting in touch.
“The whole of Dunfermline Football Club, they’re all concerned. They’ve all been praying he gets better and that’s he now on the road to recovery.
“When he came to the club last week everyone stood up and clapped. He was a top player for the club – setting lots of new records.
“We hope he’ll make a full recovery, but first and foremost he needs to just get strong again. Everybody’s thoughts are with him.
“In football terms he’s a very special person. His achievements throughout history are just amazing and his knowledge of football is incredible.
“I interviewed him for 20 minutes last week and he was absolutely fantastic. The stories are amazing.
“If you look at his record even at Aberdeen alone. If that was start to finish, it’s still astounding. But then he went on to even greater things in Manchester.
“He’s not just a legend, he’s a super-legend.”