Woods was tipped to pass the record set by his fellow American earlier in his career but he has not tasted success in one of the four major tournaments since winning the US Open in 2008.
Palmer believes that time could be running out for the world number one, who has struggled with a number of injury problems in recent seasons in pursuit of his 15th win in the Majors.
"It is going to be really tough for him now in his quest to do what Jack did," the 84-year-old told reporters. "I don't think 38 is the ultimate stopping point but it does lessen the possibility of him catching Jack.
"It's going to be tough to keep the concentration and the type of game necessary to win Majors.
"Then you've got all these young players, who are really tough and strong, who are going to make it very difficult for anyone to win a bunch of majors like Jack or Tiger did."
Nicklaus won the last of his major tournaments as a 46-year-old at the Masters in 1986.