There is a "strong case" for strengthening current football club ownership rules, the Football Supporters' Federation (FSF) says.
Questions have been raised in the leaked Paradise Papers about who controls Everton FC and whether Premier League rules have been broken.
Farhad Moshiri sold his Arsenal stake in 2016 to buy nearly 50% of Everton.
"It is essential that there is seen to be complete separation of ownership between clubs," the FSF said.
"There can be no possibility of any influence by the owners of one club over the affairs of another club playing in the same competition," said FSF chair Malcolm Clarke.
"There is a strong case for ensuring that people who have an ongoing and significant business and financial relationship outside football cannot both have a significant stake in football clubs which compete against each other."
Premier League rules state an individual who owns a stake of 10% or more in one club cannot hold a single share in another, to avoid any conflict of interest, including in games between the clubs and in transfers.
Leaks reported by the BBC's Panorama suggest Moshiri's original Arsenal stake was funded by a "gift" from oligarch Alisher Usmanov, who owns 30.4% of Arsenal, raising the question of whether his money is now in Everton.
Moshiri strenuously denied the money was a gift.
Lawyers acting for him in the Everton deal said any allegation Premier League rules had been violated were wholly false.
They say Moshiri is independently wealthy and funded the football investments himself.
Usmanov's legal representatives said there were errors in the allegations and that the investigation was a gross intrusion into their client's privacy.
'Supporters should be able to see exactly what the situation is at clubs'
Clarke added: "We know that the rules are there very precisely about shareholding, but I think there is probably a strong case for looking at the rule to make it not possible for somebody to own a football club if they still have a significant and ongoing business relationship with somebody who has got a significant stake in another club.
"There's plenty of other regulatory environments where there are very strict tests to ensure total independence and separation. I think that's what football should be looking at now.
"The supporters and the wider public often simply don't know who owns what, where it's based, who has what significant power. It isn't just a matter of Everton and Arsenal, it's a matter of football as a whole."