The Los Angeles Galaxy midfielder won just one Primera Division title and one Supercopa in four seasons with the Spanish club, while Mourinho saw his side beaten to all major silverware - apart from the Copa del Rey - by Barcelona in his first season at the Santiago Bernabeu.
But Beckham believes the Portuguese can end Barca's dominance both domestically and internationally during the new campaign.
"They have one of the best coaches in the world. Mourinho is a genius and he has proved that with all the teams he has coached," the former England captain told Marca .
"Madrid are building a powerful side with important players and they can win all the competitions.
"They are a team that strikes fear (into their rivals) and I have no doubt they will be able to win everything."
Beckham missed the opportunity to play for the Galaxy against his former club last summer as he was sidelined with an Achilles injury which saw him miss the World Cup in South Africa.
But the Londoner says he is relishing the prospect of meeting Madrid on the Spanish side's current tour of the United States, despite a shoulder problem.
"I have good memories of my time with Madrid," he said.
"I wasn't able to play last year, but I don't want to miss it this time - it will be a special game.
"I have a shoulder problem, but it feels better all the time and I have a week to prepare for the game against Madrid."
"Ronaldo is a phenomenon and one of the best players (in the world). I look forward to seeing him," he said.
Beckham was signed for Madrid by Florentino Perez in 2003, but the construction magnate stepped down two years later before taking over again in 2009, when the Englishman had already moved on.
"If Florentino had stayed, I am sure I would have retired at Real Madrid," Beckham said.