Cora called it "very difficult" to give them the news. He said that while the two will not be on the roster, they will remain with the team.
"They're gonna keep staying in shape and throwing the ball because you never know what happens," Cora said.
Both pitchers have appeared as relievers and spot starters. The right-handed Velazquez finished the season 7-2 with a 3.18 ERA. The lefty Johnson was 4-5 with a 4.17 ERA.
He said that Sale, who missed six weeks in the second half of the season with shoulder inflammation and did not have his usual velocity when he returned, "feels better." Cora said the problem was mechanics, not an injury.
"It's not a huge adjustment," Cora said. "It should be easy to fix." Sale threw on flat ground Monday and was scheduled to throw indoors in a cage on Tuesday.
"I'm 100 percent sure he's healthy and he's gonna be fine," the manager said.
Cora still has a couple of decisions to make as to which pitchers will make the final roster, which is due Friday. He said those decisions will depend on who wins Wednesday's wild-card game between the A's and Yankees.
He said that Nathan Eovaldi will be available out of the bullpen for Game 1 and will probably start Game 4 if the Red Sox face the Yankees. If it's the A's, the Sox are still deciding between Eovaldi and Eduardo Rodriguez to start Game 4.
Cora also said that infielder Brandon Phillips will be sent home. Utilityman Tzu-Wei Lin, outfielder Sam Travis and left-hander Robby Scott will be sent to the Red Sox's training camp in Fort Myers, Florida, to remain ready in case they are needed later in the postseason.