Brown lost his consultancy job at Derby when Davies left the Premier League strugglers.
And he believes the Scottish Football Association would be hard-pressed to find a better replacement for Birmingham-bound McLeish than the former Motherwell manager.
"I'm not in a position to advise but, if you were asking me, I would say Billy Davies is hard to beat," Brown told Sky Sports News. "He is available, he's outstanding and he would make a first-class Scotland manager.
"He can analyse a game exceptionally well, he has a photographic memory for analysing a game at half-time and at the end of a match.
"He has all the attributes for a top manager.
"Billy did exceptionally well to get Derby promoted and maybe he was a victim of his own success.
"I'm sure given the transfer window coming up, he would have secured Premier League status next season for Derby.
"Billy is now available and what an acquisition he would be."
But he feels the lack of day-to-day interaction with players that McLeish missed will not necessarily put Davies off.
"It may worry him because he is such an enthusiast, but I'm sure he would get immersed in coaching courses in Scotland," said Brown, the last man to lead Scotland to a major finals.
"I was eight and a half years as Scotland manager but I had plenty of work to do, I attended all the courses that they ran and I was fully occupied.
"I'm sure Billy, because he is such an industrious guy, would want not to be loafing around.
"He would go abroad, he would find out all there is to know about the game he loves, about the players in the opposition and potential Scotland players.
"I don't think time would weigh heavily in his hands."
Brown also added Joe Jordan and Roy Aitken to a list of potential successors put to him.
And he added: "There are so many suitable candidates. Scotland won't lack for quality when it comes to a new manager."