Tottenham striker Kane was off the pace during England's 1-0 win over Croatia in their Group D opener, and was withdrawn late on during that game.
The Spurs star - who is determined to leave the club this summer - suffered the same fate during Friday night's goalless draw against Scotland at Wembley, though this time he was withdrawn with 15 minutes to go while England were searching for a winner.
Kane is wanted by Manchester City, Manchester United and Chelsea, though England boss Gareth Southgate suggested that any transfer talks regarding players in his England squad would have to wait until after the Euros are finish.
Kane has repeatedly stated he is focused on England, but former Old Trafford captain Keane feels Kane is off the pace - and the transfer talk could be affecting him.
"Before the competition, there was all talk he might be leaving Spurs and maybe he has been distracted by that because that can be an energy-sapper if you think you're moving club or you're in talks with other clubs - that is not helpful," Keane told ITV.
"If England really want to compete in this competition, you need your star men to perform.
"We have seen Romelu Lukaku, Kevin De Bruyne turned up, Cristiano Ronaldo, they need Kane at his best and I am glad Gareth took him off.
"There is this talk about he is undroppable, he's the main man, if he is not performing you get him off."