Both the Scottish Police Federation and anti-sectarian charity Nil By Mouth have made the suggestion, after a street attack in Glasgow city centre, which left a man fighting for his life in hospital.
Football fan Francis McCann, 47, was reportedly stabbed on Sunday evening after the heated derby at Celtic Park, where Celtic won 2-1.
The violence, which saw two other men aged 29 and 30, also hospitalised, was described as the worst in 20 years.
But hopes are high that if the Old Firm matches were moved to weekdays, fans would have to control their behaviour - and their drinking - or else go to work with a hangover.
Charity director David Scott said: “We’ve had a terrible 48 hours after the game - one of the most extremely serious incidents that we’ve had in a long time.
“It used to be that the games were changed to earlier on a Sunday because people would be getting tanked up before the games - but now they are getting tanked up after the games instead.
“If they had a 7.45pm kick-off on a Monday or a Tuesday, that way people have to go to their work before it and their work afterwards.
“It’s a tricky one.
“What went on the other evening happened outside the club so we can’t punish a football club for something that went on outside their grounds.
“What we have to do is take a step back from the events of this weekend, give it a week or so to see what’s happened, let’s hope that the man involved pulls through.
“There should be a fixture split in a week’s time and it’s not unusual for fixtures to be changed to suit television schedules and the sponsorship there.
“We need to make sure they fit public safety.
“My sense is that if a game was on a Monday or Tuesday night at 7.45pm, then that would be a lot tighter for people who are working.
“There wouldn’t be the scope to take alcohol before - or to face the consequences the next day of work with a hangover.”
The charity was established following the murder of football fan Mark Scott in 1996.
Vice-chairman of the Scottish Police Federation, David Hamilton, said: “From a police officer’s perspective, policing any event, particularly a high-energy football match like an Old Firm match, we would prefer people to sober rather than under the influence.
“So anything that makes it easier to the police and safer to the police, has got to be a good thing.
“If that means moving it back to a Monday or Tuesday then we would welcome that.
“This is the latest in a sequence of football disorder, and it’s not just the Old Firm, it’s across the country.
“Some fans think they can behave like criminals and that it’s okay to behave like that.
“It’s not acceptable in 2019.”