The world number eight split with Ivan Lendl, another multiple winner on the ATP Tour, in March after asuccessful two-year partnership.
McEnroe also has a number of other commitments away from the circuit, but this week the American indicated he would be interested in taking on the Scot.
And Murray admits he cannot ignore the claims of a man who won three Wimbledon titles and four US Opens.
"I speak to him a lot around tournaments. I have always got on well with him and he has always said nice things to me and about me," Murray told BBC Sport.
"He obviously has a lot to offer. It is interesting - whether anything comes from it, who knows?
"Obviously when someone like that shows an interest, you have to take that into consideration because he is a great tennis player.
"He loves the game, is very passionate about it and was extremely competitive as well.
"When you are very competitive as a player, you are likely to be so as a coach as well and that is also of benefit."
Murray had hoped to have a new coach in place by the French Open, which begins on May 25, but accepts that timescale may be delayed by another month.
"I might have to wait a bit,. It also depends in the next couple of weeks because, if I do well, I'm not going to have that much time, while if I don't do well I'll have more time to chat to people.
"I would still hope definitely before Wimbledon, but I would like someone before the French Open."
Earlier this week, when McEnroe was asked whether he would be interested in coaching Murray he said: "Yes, of course.
"Some of these situations are tempting for anyone. My phone has not been ringing off the hook but, at the same time, if the right situation came along...
"I am pretty busy doing my own thing, but when you get opportunities to coach great players it is definitely something that would be tempting to think about.
"I don't think Andy can simply pick up and turn to another person that easily - that is probably why he is not with another person just now.
"He will find someone he is willing to take that chance with and there are plenty of people that would love to do it.
"It's a great job to be able to work with someone as good as him.
"I am sure there are plenty of people applying for the job. It is just a matter of him settling down and deciding where he wants to go."