The Tottenham Hotspur striker, the Premier League's leading scorer last season, failed to shine in England's opening match and, according to Shearer, lacked energy.
Leicester City's title-winning striker Vardy had to kick his heels on the bench in Marseille where England were held 1-1.
"(England coach) Roy Hodgson would be within his rights to tell Harry Kane that his performance in Saturday's draw against Russia was not good enough and he has to improve or he will change things up front -- and rightly so," Shearer told the BBC.
"I am a huge fan of Kane, and he has had another fantastic season for Tottenham, but he looked very leggy and tired against Russia on Saturday."
Shearer still expects Kane to lead England's line against Wales, but says he needs to step up.
"I would not alter anything about England's starting XI or the system they played in Marseille for their next game against Wales," he said.
"I would tell Kane -- and also Raheem Sterling, whose final ball was yet again very poor -- that, although I am going to stick with you, if you do not do better then I will make those changes during the Wales game. It is up to them to respond."
Kane played as the central prong in a three-man attack against Russia with Adam Lallana impressive on the right and Sterling less so on the left despite plenty of the ball.
Kane only touched the ball once in Russia's penalty area.
"Just because you have one bad game does not mean you are a bad player and Kane will be looking to Hodgson to give him another opportunity," Shearer said.
"But Hodgson has a lot of other strong options, particularly in the centre-forward positions."
Shearer said he had been impressed with England's midfield in which captain Wayne Rooney often dropped deep to allow Dele Alli the license to roam forward but still could have scored a couple of goals.
"He played the new position very well and combined well with Eric Dier and Dele Alli too," Shearer said.
"Dier sat in but Rooney and Alli had to decide when to stay and go forward -- and they got it right."
(Reporting by Martyn Herman; Editing by Neil Robinson)
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