Speaking Tuesday before the Dodgers' 4-1 defeat to the Chicago White Sox, McGwire said, "There could a triple crown. I see some MVPs."
When Puig made his major league debut exactly one year earlier, McGwire was convinced the Cuban outfielder was a special player. What McGwire has seen since then has only reaffirmed that belief.
"I've only really seen one person with that kind of talent and it's Albert," McGwire said.
McGwire was talking about Albert Pujols, whom he played alongside and coached with the St. Louis Cardinals.
Puig and Pujols are two of only five players in baseball history to have collected at least 190 hits and 30 home runs within a year of their first major league game. The only other player to do that in the last 80 years was admitted drug cheat Ryan Braun.
"These type of talented baseball players come around only so often," McGwire said. "Albert came, then [Mike] Trout came, then Puig came."
But McGwire said there's one significant difference between Pujols and Puig. Pujols was already a disciplined hitter when he broke into the major leagues.
Puig had to turn himself into one.
Asked if he had ever seen a hitter make as rapid a transformation as Puig, McGwire replied, "That's a great question."
McGwire pondered the question for several moments, then replied, "I don't think I have."
By the end of last season, opposing pitchers appeared as if they had figured out Puig. Pounded inside with fastballs and tricked into chasing breaking balls off the plate, he batted.214 in September. He looked completely lost in the National League Championship Series, hitting .227.
Puig isn't chasing nearly as much this season.
He has walked 27 times in 54 games. He drew only 36 walks in 104 games last year.
The improvement is a testament to Puig's intelligence, according to McGwire, who said Puig is as smart as he is physically gifted.
"He's really smart," McGwire said. "People see the antics he does and the fun he has, but he's a lot smarter than people think he is.
"You have to give him a lot of credit. He's made some unbelievable adjustments."
Puig speaks about the adjustments he had made in general terms. He talks about his cage work and the advice he has received from teammates such as Adrian Gonzalez, but offers few details.
McGwire was more forthcoming, saying that whereas Puig tried to cover the entire plate last season, he now looks for pitches on one half.
"It's very rare that a guy hits 17 inches of the plate," McGwire said. "You have to pick 81/2. Now, those adjustments switch throughout the game, depending on how the guy's going to pitch you."
The plan is working. Puig is batting .343 with 11 home runs and 40 runs batted in. He is second in the National League in batting, tied for seventh in home runs and tied for second in RBIs.
Because opposing pitchers are now armed with detailed scouting reports, McGwire considers what Puig is doing this year more impressive than he did last year.
"When he got off to a great start, they were just challenging him," McGwire said. "Then they found some things they could really get him out on. Going into this year, he's done a tremendous job of narrowing those things down."
Puig is also demonstrated increased restraint on defense, leading Manager Don Mattingly to say he thinks Puig is now the best right fielder in baseball.
"Still with all the energy, but less out of control," Mattingly said.
Fans continue to be drawn to Puig. When the results of the latest round of All-Star balloting were revealed Tuesday, Puig was leading all National League outfielders in votes.
McGwire said this is only a start. As surprisingly polished as Puig has looked at times, McGwire said he is still in the process of learning the game.
"Really, the sky's the limit for what he can do for the rest of his career," McGwire said. "You're that talented and you're that smart, you can pretty much take over the game if you want.