United have impressed in pre-season, reaching the final of the International Champions Cup in the United States after a 3-1 victory over Real Madrid on Saturday.
That was the latest sign of improvement under Van Gaal, who officially replaced David Moyes at Old Trafford last month after leading Netherlands to the semi-finals of the World Cup in Brazil.
The Dutchman has already stamped his authority on the squad ahead of the 2014-15 season, with United looking to respond to a disappointing seventh-place finish in the Premier League last term that saw them miss out on European football.
And Fletcher believes they are already showing signs of being able to challenge with the best sides again under Van Gaal.
"He [Van Gaal] demands a lot, but the players are responding to to it and enjoying playing in the new system and playing under him," said the Scotland international.
"We've got a fantastic team spirit - we were talking about it last night amongst ourselves.
"We're all close and we're all pulling in the right direction and we all want the best for Manchester United.
"We're disappointed with where we finished in the league [last season] and as a a result of that Liverpool, Man City, Chelsea, they all finished above of us.
"We're disappointed but next season we want to show what we're capable of and hopefully we can can do that."
That is a sentiment echoed by De Gea, with the Spain goalkeeper revealing that Van Gaal's arrival has already had a significant impact on the mentality within the squad.
"He [van Gaal] is hard when he has to be. Really, he has surprised me," said De Gea.
"He is a great head coach and he is providing the team with a very good character. You can see it so far. I think we are totally different from last season.
"I think a lot of the big teams have been reinforced very well.
"We have also have been reinforced, with some players like Ander Herrera, and Luke Shaw on the left side.
"Right now, it could well be that we have one of the best squads."
