The Mets also will likely be without first baseman Lucas Duda (left elbow) and backup first baseman Wilmer Flores (knee infection) Friday.
Cespedes will have an MRI on Friday, and manager Terry Collins called the injury "a concern".
The severity of the injury was originally unknown, but the Mets eventually announced that Cespedes was experiencing cramping in his left hamstring.
Zimmerman batted only.242 with 36 homers and 157 RBIs over the previous three seasons, a stretch in which he was limited to 271 games by a variety of injuries.
Jared Diamond of the Wall Street Journal noted the Mets are saying Cespedes has a cramp in his left hamstring.
Olympic hurdles champion Rollins handed one-year ban
Her gold medal, part of a medals sweep by USA 100-meter hurdlers at the Rio Games previous year , will not be stripped. The arbitrators added that Rollins showed "no evidence of avoiding testing, masking drug use, or using drugs".
As his absence remains a fact, Jay Bruce will be responsible for first base, while lefty Sean Gilmartin was recalled back up from the minors to take his place in the roster.
Cespedes limped out of Thursday night's loss to Philadelphia in the fifth inning.
Céspedes underwent an MRI on Friday, but the results haven't been released yet, according to Mike Puma of the New York Post.
"We have Monday off, but I think unless he makes a miraculous recovery today, we'd be foolish to push him with the extra day on Monday", Collins said. He is 2-2 with a 3.12 ERA in seven starts against the Nationals.
DeGrom doesn't have a decision in three starts this season despite a 1.89 ERA. Catcher Travis d'Arnaud, who was limited to pinch-hitting duties Thursday because of a right wrist contusion, could return to the lineup. David Wright can't be counted on at this stage of his career, playing in just 75 games total the previous two years; the club placed him on the 60-day disabled list Thursday. It is still early in the season, but the Mets can not afford to fall into an early hole.