If the former England captain accepts the Leith post, he will become the club's ninth manager since 2001.
Fenlon resigned on Friday just two days after administration-hit Hearts racked up their second shock Edinburgh derby win of the season.
But Riley, chairman of the Hibs Supporters Association, claims it is the man at the top who the capital club's faithful are running out of patience with.
He told Press Association Sport: "The fact we have been through so many managers in the last few years is a big concern to everyone who supports Hibs.
"The managers can't all be wrong and somebody has to take responsibility for that. Perhaps we need a director of football.
"I think we're already at the point where if Petrie doesn't get it right with Butcher there will be an almighty outcry.
"But we would rather that he got it right this time. Rather that than having to change the manager again or we end up getting relegated. We just want someone to come in and get the players working."
Riley led the calls for Fenlon to be replaced as boss after the 1-0 League Cup quarter-final defeat to the Jambos.
The Hibs fans association leader labelled the defeat a 'disgrace' as he said enough was enough following earlier embarrassments under the Irishman, which included losing last year's Scottish Cup final 5-1 to the Tynecastle side and the 7-0 Europa League humiliation at the hands of Malmo.
But he gave Caley Thistle boss Butcher - who is still to put pen to paper on a deal with Hibs after the Highlanders granted him permission to discuss a move on Tuesday - a cautious welcome.
"Terry Butcher is - alongside Stuart McCall - the leading manager in Scotland right now," said Riley.
"They are the two that have done well recently, along with Derek McInnes, although I don't think we would have much chance of getting him from Aberdeen.
"But there is no way of saying for sure if Terry Butcher is the right man or not right now. He will have faith in his ability but we just can't be sure.
"I remember a few years back we signed who everyone thought was the best manager in the league at that time and that was Bobby Williamson. He ended up being a disaster. His famous quote was: 'If you want entertainment, go to the cinema'.
"But at least with Butcher, he seems to play an attacking game and knows how to pick out a player - and Inverness certainly seem to beat us all the time. So if he can reverse that trend that would be progress."
Riley added: "Inverness under Terry have done well. They are flying high in the league and have reached a semi-final of the League Cup for the second year in a row.
"But it is difficult to say if he is the man to take us forward. I'll be worried if he turns the job down though, let me put it that way."
Fenlon did lead Hibs to an unbeaten campaign against Hearts last term - the first time the Easter Road club had achieved such a feat since 2001 - but could do nothing as his side's financially-stricken city rivals won both of this season's derbies.
Now Riley says if Butcher does land the job, the minimum he must do is restore the Edinburgh balance.
"With the pool of players we have got, we should be able to finish in the top six," he said. "There is no doubt in that.
"As for the derbies? We always need to win them. Anyone who says they are just another game isn't a Hibs or a Hearts supporters. It's never just another game.
"So we would expect him to win the last two or three derby games, depending how many are played this year. The bragging rights are important from these games and it would good to get the upper hand for once."
- Sports & Recreation
- Terry Butcher
- Pat Fenlon