The crime author, who led the protest against Raith Rovers’ signing of David Goodwillie and withdrew sponsorship from the side, said there should be “some kind of independent body that regulates the whole game”.
Raith Rovers said Goodwillie would not play for them after a public outcry following his signing from Clyde.
Mr Goodwillie was judged to be a rapist in a civil case against him in 2017, with the victim stating it was “shameful” he was allowed to continue playing football following the case, which was the first of its kind in Scotland.
Ms Sturgeon said as well as any move to introduce a regulator, a requirement to undertake a version of the “fit and proper person” test that is used for club owners could also be brought in, to ensure players are role models for younger fans.
Speaking during First Minister’s Questions on Thursday, Ms Sturgeon said the football authorities should look at the issue around regulation.
She said: “Firstly I do think that Val McDermid’s proposal merits further consideration. In the interests of full transparency, Val McDermid is a friend of mine, but not withstanding that I do think she has said many sensible things on this issue.
"Football clubs have a particular responsibility which reflects their special place within our society.
"Being a footballer – and for different reasons obviously, but I suppose a bit like being a politician – it is not an ordinary job.
"People look up to footballers. That’s perhaps not always true of politicians, but there is a responsibility on football clubs to make sure that those playing for them are role models for the wee boys and the wee girls who do look up to them and see them as heroes.
"I think the football authorities perhaps need to reflect on recent events and ask the question about whether current rules and regulations are sufficient.”