“Lifestyle clinics” are among a range of other proposed services, with local doctors likely to refer patients they feel could benefit.
The Evening News understands club bosses are considering physical alterations at Easter Road to accommodate the centre.
However, it is not yet known whether this would involve building an extension.
Hibernian Community Foundation, NHS Lothian and scores of partner groups will help deliver the plans as part of the GameChanger project.
Club bosses said their first priority would always be the side’s on-pitch performances, but stressed they had a vital role to play in the general wellbeing of local communities.
Chief executive Leeann Dempster, pictured left, said: “Consider this statistic – one in 48 people living in Scotland pass through their club turnstiles every match day.
“That’s an incredible number.
“Then add to that the acres of daily column inches, hours of weekly broadcast coverage, and endless digital media coverage that football spawns.
“What does this mean – apart from the fact that we are, genuinely, ‘fitba’ daft’?
“It means our sport has a unique opportunity to reach out to people and communities in a way that nothing else can.
“And at the moment, we don’t do enough with that potential. At Hibernian, we want to change that.”
She added: “This club, which has pioneered so much in the game, now wants to be pioneering in our wider society.
“That is why we are all so excited about our new GameChanger partnership.”
Hibs leaders said GameChanger would be supported by organisations such as local authorities, universities, colleges, charities private companies and individuals.
They added that the scale of the work proposed was greater than anything currently being undertaken within football.
News of the plan has sparked interest among ministers and senior MSPs, including Iain Gray, who chairs Hibernian Community Foundation, Kenny MacAskill, Keith Brown, Malcolm Chisholm and Kezia Dugdale.
City doctors have also expressed support.
Dr Richard Williams, a non-executive director of NHS Lothian, said: “As a local GP I welcome the enormous opportunities that GameChanger offers to improve the health and wellbeing of our citizens.”
Hibs bosses said other projects were in various stages of planning, among them a community growing scheme and “state of the art” learning and innovation facilities.