Chelsea say Petr Cech has regained full consciousness after undergoing surgery on his fractured skull.
Keeper Cech, 24, was hurt in the first minute of Chelsea's win over Reading after colliding with Stephen Hunt and could miss the rest of the season.
A Chelsea statement said Cech's condition is improving and that he has been visited by family and team-mates.
However, Cech's father said his son will be sidelined for a year and called for Hunt to receive a lengthy ban.
"This was no accident," said Cech Snr told The Sun newspaper. "Hunt hit him on purpose and Petr will be out for a year.
"[Manchester City's] Ben Thatcher received an eight-week ban [for a challenge on Portsmouth's Pedro Mendez] but I would be harsher than that.
Hunt has denied deliberately injuring Cech but Blues boss Jose Mourinho has asked the FA to review the incident.
"If Petr Cech returns too soon from his head injury, the result could be fatal"
Hunt said: "I can guarantee that I did not attempt to injure Petr and I am very upset that the collision has resulted in such a bad injury.
"When Petr slid in to collect the ball, our momentum meant that a collision was unavoidable.
"I think the TV pictures show that the collision was a complete accident, and I'd like to think that my team-mates and opponents throughout my career in football would say that I'm not the kind of person to deliberately hurt an opponent.
"Serious injury is the worst part of football and I feel terribly sorry for Petr. I have written to him to wish him a full recovery."
Reading boss Steve Coppell added: "We support Stephen 100%. He is a fair and honest player, and anybody who knows him would agree that he is not the kind of person to hurt Petr Cech on purpose.
"We hope Petr makes a full recovery and is back playing soon."
Cech underwent a successful operation at the Radcliffe Infirmary's specialist neuro-surgical unit in Oxford.
"Mike Riley has told Keith Hackett that he did not request Petr Cech to leave the field of play"
Fellow keeper Carlo Cudicini was also carried off during the match after a collision with Reading's Ibrahima Sonko and Chelsea have not commented on how long either player will be sidelined.
"I would think Cech is likely to be out for at least six months," Peter Hamlyn, consultant neuro-surgeon and director for the centre of sports and exercise medicine at St Bartholomew's Hospital told the Daily Telegraph.
"That is about the length of time it takes for stout bone healing to take place. If the brain underlying the fracture has been greatly injured, his time out may be even longer.
"If Petr Cech returns too soon from his head injury, the result could be fatal."
Mourinho has called for action to be taken over the challenges of Hunt and Sonko but the Football Association is yet to receive a formal complaint from Chelsea.
Meanwhile referee Mike Riley has rejected claims he ordered Cech to leave the field for treatment.
Cech did crawl towards the touchline but Riley has told referees chief Keith Hackett he followed the correct protocol for head injuries.