Peter Lawwell says the new Scottish FA chief executive needs to build bridges with the country's clubs.
Stewart Regan left the role last week while the Scotland manager's post remains vacant.
Celtic chief executive Lawwell was previously on the SFA board and has said the organisation needs restructuring.
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"What we need is somebody that will bring Scottish football together," said Lawwell.
"They need to get more control. They need to be delegated the authority to do that but they also need to be guided by the good people that are on the board there - the non-execs and the club reps.
"I'm sure there's a lot of experience there, there's a lot of ability there and whoever comes in will need that support to take the association forward.
"I have no authority. I'm not on the board, I'm not on any committee that's there to select the chief executive. I'm remote from that.
"Clearly, we're a big club in Scotland and the SFA or whoever's around there, it would be wise for them to have the opinion not just of Celtic but other big clubs within Scotland in terms of where the direction is."
Lawwell was asked if he would want Scottish Professional Football League chief executive Neil Doncaster to replace Regan at the SFA.
"Neil is a very able guy," said Lawwell.
"Under pretty difficult circumstances over the years at the SPFL, he's done a very good job but I would prefer Neil Doncaster on that basis to remain at the SPFL and continue that work because there are challenges ahead."
SFA's relationship with the clubs 'as bad as it's been'
Regan's departure followed the SFA's unsuccessful attempt to recruit Northern Ireland boss Michael O'Neill as Scotland manager and controversial friendly matches against Peru and Mexico planned for June near the time Scottish clubs will have European qualifying fixtures.
"It's not great at the moment," Lawwell said of the SFA's relationship with Scottish clubs.
"It's probably in terms of my time in the game as bad as it's been and that for the board of the SFA, in my view, and the chief exec would be a big priority - to build those bridges again.
"We probably need somebody who really does understand that distinction between the SFA and the SPFL and the clubs and the regulator and the governing body. One way of describing it is the private sector and the public sector - who understands that difference."
Lawwell says the June international friendlies were "certainly a factor that demonstrated the gap between the clubs and the SFA".
"We found out about it for the first time reading it in the paper and clearly we've had eight [Celtic] players in the [Scotland] team so in terms of building relations with clubs, it certainly wasn't the best way to go about it," he explained.
However, he said of the future: "I am hopeful because there's only one way to go and you've got good people around and the fact that there is now a realisation that things haven't been perfect and they need to change - and that's from the media as well.
"Some of the stuff I'm reading, probably all the stakeholders, I'm sure, will be thinking the same thing clubs are. I'm unsure of the view of the regional associations or regional authorities but now there is a realisation that we have an opportunity to get things right."
Dundee manager and former Scotland winger Neil McCann has suggested the SFA should appoint a new Scotland manager before recruiting a chief executive but Lawwell said: "Ideally, if you put it in a club context, if a chief exec arrived shortly after the manager then it wouldn't be ideal.
"For me, the priority is the chief executive at the moment because there is a gap and [interim chief executive] Andrew McKinlay's decision to move in two or three months' time [to Scottish Golf] makes it a bigger gap so I would think that the priority is to get someone in quickly, get hold of the reins and then appoint the manager.
"Again I have no part to play in this, but if there is a preferred candidate there who the board feel that they may lose then that might change."