Murray recorded the biggest triumph of his career with a comprehensive victory over Roger Federer to take gold at Wimbledon but McEnroe believes that the British number one has to target a win at the US Open.
The seven-time Grand Slam winner said: "Winning the Olympics doesn't really get the monkey off Murray's back.
"Obviously he is going to come in as the number three seed, which doesn't necessarily change a whole lot for him because, if things go according to form, he'd still have to beat two out of the top three guys.
"Nadal's out of the picture but Murray's beaten Nadal at the US Open before, so it's not as if he didn't feel he could do that in the past. He's still got the two guys that he's lost to in the Grand Slams to deal with."
Former Wimbledon champion Goran Ivanisevic recently tipped Murray to win at Flushing Meadows next month.