Wilk, 29, is 2-3 with a 5.91 ERA and 1.41 WHIP in six Triple-A starts this season. He has made nine total MLB appearances.
Syndergaard, 24, was diagnosed with a torn lat in his throwing arm last month. The 2016 All-Star is expected to be out until at least the All-Star break.
"Realistically, it's going to take a while. We don't know when he's going to be back," general manager Sandy Alderson told the Associated Press (via ESPN.com). "It's going to be much later in the season."
Syndergaard first experienced discomfort in his throwing arm before an April 27 start against the Atlanta Braves. He was scratched from that game but returned for 1.1 innings in an April 30 start against the Washington Nationals. Alderson said Syndergaard refused to undergo an MRI when he first experienced the discomfort.
"He's definitely in that case, assuming that we acquiesce, eliminating some information that probably would have been useful, but that doesn't mean that it would have been dispositive," Alderson said. "The situation was such that to me the MRI was not a critical element to the decision-making."
Syndergaard was 1-2 with a 3.29 ERA and 1.10 WHIP at the time of his injury. The Mets are off to a disappointing 14-15 start after being viewed as a potential pennant winner in spring training.