The Manchester City shot-stopper enjoyed a successful domestic season, helping Roberto Mancini's men pip Schmeichel's former club to the Premier League title on goal difference.
However, the 48-year-old, who famously helped Denmark to an unlikely triumph at Euro 1992, says Hart will have to improve further in a Three Lions shirt.
Schmeichel told the Daily Mail: “Joe Hart is incredibly talented and has done very well since coming into the Premier League.
“But you just have to look back in history to see how difficult it can be to be England’s goalkeeper in a tournament.
“It’s the biggest thing he’s done in his career up till now. If he can maintain the level that he’s shown so far it will make him.
“But for any England goalkeeper, a European Championships or World Cup can break them.
“Goalkeepers make mistakes all the time and more often than not nobody notices, but they see the mistakes that cost you goals. Everyone knows what happened to Robert Green [at the World Cup in 2010]. It’s taken him a long time to recover. You have to remove those mistakes from your game.”
Schmeichel went on to hail Hart's talent and insisted that the absence of key players for England means this could be his chance to shine on the biggest stage.
“This is the right moment for him to take the next step,” he added.
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“Manchester City disappointed in the Champions League which is another big stage so we haven’t had the opportunity to see him play in those games but that chance will come again.
“He sees the danger before it happens which is an incredible talent to have. He inspires confidence in his defenders and is fantastic distributor of the ball. No team can compete as City have without a goalkeeper who can actually play football.
“They pass the ball around so quickly and sometimes get in trouble but by passing the ball back to the goalkeeper, he can start another move. If you don’t have that ability, you lose valuable seconds where you have opportunities to change the game.
“In the absence of Wayne Rooney, Frank Lampard and Gary Cahill, England need their big players to play well and Joe is one of them. He is a born leader and has to bring his A-game.”