The 27-year-old, who has so far declined to appoint a headline coach to replace Ivan Lendl, despite an invitation from McEnroe to discuss a proposed partnership, has never won a title on clay, but put in one of his finest performances on the surface, albeit in a losing effort, against Rafael Nadal in the quarter-finals in Rome.
John, in addition to being one of his brother's fellow pundits with the US broadcaster ESPN, has his own tennis academy in New York.
"I know they have a very good relationship and I think John, in the right situation, could be a great coach," said Patrick McEnroe. "The question is whether he would be willing to spend the time that it would take, depending on what Andy is looking for. Andy has got a solid team around him; Dani [Vallverdu] has been around him for ever. As for where his coaching situation ends up, that is the $64,000 question, but, from what I have heard Andy say, it sounds like he is looking for someone who is going to be with him for quite a long period of time, a long-term solution, and not a quick, part-time band aid."
Murray has never gone beyond the semi-finals at Roland Garros, a tournament in which opportunities are distinctly limited because of the perennial dominance of Rafael Nadal. Impressive as Murray was in Rome, Patrick McEnroe believes his expectations should be modest.
"I don't expect him to be a real threat to win but I know he will want to play well there, as he did in Rome, because the best part of his year normally is the grass - defending his Wimbledon title - then the US summer on the hard court," said McEnroe. "A good result for him I think would be to make the second week, because I feel he is vulnerable to a lot more players on clay than he is on any other surface.
"I believe he is going to use the French Open as a way to get him going for Wimbledon, get his fitness up, get his back right, and get his confidence going. The pressure is off in some ways at Wimbledon but I am sure that walking out at 1pm for his first match will be very emotional both for him and the crowd. He will be primed to peak there."
Murray's fellow Scot, Jamie Baker, meanwhile, is backing the Dunblane man to take advantage of the "most open French Open in years" and enjoy a strong run at Roland Garros.
Baker, who has swapped his tennis clothes for a suit and tie and a job in a bank since quitting the sport last summer, has known Murray since their junior days, and believes he will be hitting full stride by the time Wimbledon comes round. "If he [Murray] gets a decent draw in the first couple of matches and plays his way in, he's quite fresh," said Baker. "He played so well against Nadal the other night, I genuinely think it's really open and a lot of it comes down to the way the draw falls.
"If you look at Andy's game, what he does well is just slightly less effective on that surface, just like what he does well is slightly more effective on a grasscourt," he said.
Baker's retirement last summer came just before Murray beat Djokovic to win the Wimbledon title and the 27-year-old believes Murray's performance in Paris will be crucial to him arriving at the All England Club in the right state of mind to make a stout defence of his title.
"I think that'll be the big thing for him," said Baker, who will be an analyst for British Eurosport's coverage of the French Open. "In his mind, I think he's still trying to get some momentum back, get the genuine confidence back after his back surgery. He's definitely not 100%; he's not doing things instinctively as he was [at Wimbledon] but he's not a million miles away. There have been some really good patches from him, just not the consistency."
Three Britons are in action in the second round of qualifying at Roland Garros today. Heather Watson and Johanna Konta both have winnable matches while James Ward takes on fiery American Ryan Harrison.