Puma added that the Mets are considering several options for their next manager, including Robin Ventura, Alex Cora, Kevin Long, Bob Geren and Chip Hale.
While the Mets reportedly have no plans to bring Collins back, Puma noted that they may allow the 68-year-old Collins to announce his retirement rather than saying they are moving on.
All of the managerial candidates Puma mentioned have ties to the Mets organization.
Ventura played third base for the Mets from 1999 through 2001. Most recently, he served as manager for the Chicago White Sox from 2012 through 2016.
Cora is currently a bench coach for the Houston Astros and spent two seasons with the Mets as a utility infielder in 2009 and 2010.
Long has been the Mets' hitting coach since 2015, and he previously acted as the New York Yankees' hitting coach from 2007 through 2014.
The 56-year-old Geren is a veteran manager and coach who managed the Oakland Athletics from 2007 through 2011. He is currently the Los Angeles Dodgers' bench coach, and he was the Mets' bench coach from 2012 until 2015.
Hale is the Athletics' current third-base coach. He served in the same capacity for the Mets in 2010 and 2011, and he was the Arizona Diamondbacks manager in 2015 and 2016.
The Mets own a record of 67-90 this season, which is good for only fourth in the National League East.
New York has reached the playoffs twice in seven seasons under Collins, including a trip to the World Series in 2015 where it lost to the Kansas City Royals in five games.