On Tuesday, Franco had said during an interview on Sirius XM Radio that the Mets, then on a five-game losing streak, were in need of a leader, and that Wright needed to step up and "be that guy."
Wright publicly shrugged off the comments, saying, "With all due respect to Johnny, he doesn't know what's going on in this clubhouse."
On Thursday, he got a voice mail from Franco, who Wright said apologized for his comments, The New York Times reported.
"Johnny and I have a great relationship," Wright told The Times. "I'm glad he called and apologized. I've always respected his opinion. But he's not in here on a day-to-day basis, so he can't really know what's going on.
"If it was one of the guys in here that said something like that, it would be one thing. But when it's someone from the outside, I really don't feel the need to defend myself. It doesn't bother me at all."
Information from The Associated Press was used in this report.