The rookie who pushes Tom Selleck's character off the Yankees roster was played by Frank Thomas, who went on to become one of the best MLB hitters in the 1990s. He won two MVP awards and was elected to the Baseball Hall of Fame in 2014.
At one point in the movie, when the pitchers refuse to pitch to Selleck's character because they don't want him to break the home run record, Selleck turns his bat the other way around and challenges pitchers to pitch to him as a sign of protest. This comes from a real-life incident a few years before the movie in the Japan Leagues when Hanshin Tigers slugger Randy Bass was on the verge of breaking the single season home run record there and pitchers refused to pitch to him. Bass similarly turned his bat upside down to protest.
Nolan Ryan (Pitcher for the Texas Rangers) agreed to pitch to Tom Selleck to give him the experience of a 100 mph baseball headed straight at him.
While preparing for this movie, Tom Selleck spent time with the Detroit Tigers during 1991 spring training. He was even brought into a game as a late inning pinch hitter, managing to foul off one pitch before striking out.