Palat left the Lightning's 5-4 overtime loss in Game 1 after playing the first shift of the third period with an upper-body injury.
Game 2 of the Stanley Cup Playoff series is set for Friday at Tampa Bay Times Forum (7 pm ET; CNBC, CBC, RDS).
"I don't know," Cooper said when asked how Palat was doing. "I guess we'll call him day-to-day, but then again aren't we all?"
Palat missed the final game of the regular season Sunday with an injury. He was hit in the head the game before by Columbus Blue Jackets defenseman Jack Johnson but never was officially diagnosed as having a concussion, and Cooper said that has been ruled out as a possibility here.
"It's upper-body related," Cooper said. "He was tested for all that stuff earlier and that's not his issue."
Cooper did not deny that Palat's injury is related to what kept him out of the final regular-season game.
"Maybe partially," Cooper said. "He's been a part of some really physical hits last few games. In all seriousness, he'll be a game-time decision.”
Cooper was asked if there's a chance Palat will play.
"There's a chance, yes," Cooper said.
The same cannot be said for Lightning goalie Ben Bishop.
Bishop skated on his own for a second straight day Thursday, taking the ice with Lightning goaltending coach Frantz Jean for a little more than half an hour in full gear. It was a change from his session Wednesday when he didn't have his chest protector on. Bishop did not take any shots, working more on his mobility with Jean.
Cooper said Bishop's return is not on the radar for now.
"I don't think we're going to see Bishop any time soon. But I don't know what soon is," Cooper said. "This series would have to extend a little bit for him to come in."
Anders Lindback replaced Bishop in Game 1 and made 39 saves in his first playoff start.
Bishop sustained the upper-body injury in a game April 8 against the Toronto Maple Leafs while diving to his left to catch a puck.