Cora said Price, who was sent back to Boston for testing, will throw a bullpen session Thursday and then the team will decide what to do next. Price was scratched from Wednesday's start against the New York Yankees.
"For me this is quote-unquote good news compared to anything else," Cora said.
Price was removed for precautionary reasons after one inning of work against the Yankees on April 11 because of numbness in his throwing hand, including the loss of feeling in his fingertips.
Cora said the problem resurfaced during a bullpen session Sunday.
Carpal tunnel syndrome is an uncommon injury for baseball players. It's caused by repeated motions, such as typing, and is more common among esports athletes. Price is a longtime video game enthusiast, and this season, Price and teammates have been logging long hours playing Fortnite; Price told The Associated Press last month the game has "kind of taken over."
Cora said Fortnite may be discussed when he sits down with Price on Thursday to lay out a recovery plan.
"From what I know, David has been playing video games his whole life," Cora said. "It seems like Fortnite now is the one everybody's playing, but last year I saw a lot of guys playing FIFA. That's all I know. There's a lot of teams playing Fortnite. But we'll talk about it."
Cora said treatment options for carpal tunnel syndrome could include icing or massages.
"We'll see what works for him," Cora said. "As you guys know, we need him. We need him here. We need him to compete. He's a big part of what we're trying to accomplish."
Price, who started the season with 14 straight scoreless innings, allowed four runs on April 11 before exiting the eventual 10-7 loss to the Yankees. He threw only 35 pitches -- the shortest start of his 11-year career.
Since then, Price has made four more starts, going 1-3 and allowing 20 runs (17 earned) in 22 innings.
Price is 2-4 with a 5.11 ERA in seven starts during his third season with Boston. The 2012 AL Cy Young Award winner signed a seven-year, $217 million contract with the Red Sox in December 2015 but has been slowed by injuries the past two years.
Information from The Associated Press was used in this report.