Both players were injured during a spring training workout Tuesday.
Hamilton struggled in his first season with the Angels after agreeing to a $125 million, five-year contract, hitting .250 with 21 homers and 79 RBIs. The 2010 AL MVP reported to spring training 20 pounds heavier than last year after losing about 25 pounds following the 2012 season.
Hamilton left the clubhouse on crutches Tuesday after getting hurt during a baserunning drill.
"It's something that will definitely heal on its own and it's just going to take some time and the timeframe we're getting is the minimum of a couple of weeks," Angels manager Mike Scioscia said Wednesday. "As of right now we're very optimistic that he'll get back in here and he can get back in shape for the start of the season."
Wilson, also starting his second season with the Angels. was hit by a liner off the bat of Yorvit Torrealba.
"Everything is fine, except that pitch I threw. Shoulda caught it," Wilson tweeted. "Lesson is to not pitch to contact!"
Wilson was bleeding slightly after being hit with the line drive and was taken in for a CT scan. He blamed himself for the mishap.
"I should have caught the ball," Wilson said, "If I ever get hit, it's my fault."
Wilson added that he did not get his usual amount of sleep the night before as he was getting "amped up" for Tuesday's batting-practice session.
Wilson, 17-7 with a 3.39 ERA last year, could make his exhibition debut Saturday.