Talks to bring Mourinho to Old Trafford have entered a second day.
Van Gaal was sacked on Monday, two days after winning the FA Cup.
"The fans at Man United demand entertaining football and I'm sure Mourinho will come and try and give it," said former midfielder Scholes.
Van Gaal led United to fifth place in the Premier League but was criticised for his lack of attacking football, with the team only scoring 49 league goals this season.
Scholes, who won 11 league titles and two Champions Leagues during a 20-year career at Old Trafford, was among Van Gaal's critics, claiming United were disjointed and played boring football under the 64-year-old.
He also questioned whether the former Netherlands, Barcelona and Bayern Munich boss really understood the "ethos of the club".
"I think Mourinho will have watched that closely and will have seen what happened," Scholes, 41, told BBC Surrey. "He's a top manager, he's proven that before and hopefully he can do that again."
Scholes also said he was optimistic former team-mate Ryan Giggs, who was assistant manager under Van Gaal, will stay at Old Trafford.
"I think it's important that there's some form of continuity," he said. "Hopefully he'll still be at the club, hopefully still working with the first team.
"He knows the club inside out, he's been there for nearly 30 years now."
The impending appointment of former Chelsea boss Mourinho has drawn a mixed reaction from former United players.
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Ex-goalkeeper Peter Schmeichel says the Portuguese "could save" the club, while Hull City boss Steve Bruce believes the opportunity to bring in the three-time Premier League winner was "just too good to miss".
But Old Trafford legend Eric Cantona told the Guardian his former club should have gone for Pep Guardiola, who will join Manchester City instead.
Cantona, who celebrated his 50th birthday this week, said he "loved" Mourinho but claimed the style of football he chooses to play is "not Manchester United".
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Jose Mourinho will learn from Louis van Gaal's experience and deliver exciting football at Manchester United should he become manager, says club legend Paul Scholes.