Dortmund are riding high on an 11-match unbeaten streak but face tough odds against the LaLiga team under scorching heat in East Rutherford, New Jersey, on Saturday, where Kovac expects a packed house with a tough crowd.
The players "know exactly what does it mean to play against Real Madrid tomorrow", he told reporters at MetLife Stadium.
"We're expecting a lot of supporters, especially for Real Madrid. We're here as an underdog but we also want to show our best face and hopefully - this is what we are believing - we can pass to the next round."
Real Madrid are unbeaten against Dortmund in their last six meetings, having won their last four, including their 2-0 win in the Champions League final last year.
The German team hope they can flip the odds after entering something of a revival since Kovac took the helm in February. The Croatian manager said his team are operating at the "highest level" of confidence against Real Madrid.
"We need high self-confidence when we face Real Madrid," he said.
"We finished the season fourth (in the Bundesliga season), but it was a photo finish, and we had tough games to play.
"We played big opponents, and the players, they showed they are able to compete with all of them. So I think this is a big reason for me to believe in my players - they know exactly what is necessary tomorrow to be on highest level."
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