The 19-year-old midfielder has been linked to a number of clubs in recent months, including several in Germany after being dubbed "U.S. Havertz" by Bild.
And Marsch is thrilled that Aaronson will continue his development in Salzburg, and he believes that the midfielder has the personality to take his game to an even higher level in the coming years.
“I'm sure it's hard for Philadelphia fans to think about losing him next year," Marsch told Extratime Radio. "But I also think it'll be a lot of fun for them to watch him play in Champions League and Europa League and then in some big matches here in Europe. We think he has a big future, not just here, but beyond."
He added: “We're really excited to have him here. I think he'll be a really good fit. The conversations I've had with Brenden have been really good. He's really mature for a 19-year-old young man, and that's something that I think is always important.
"When when you talk to these young players and when you get the feel after a 10, 15 minute conversation like you're talking to a grown man, that's always, I think, such a good indicator for the potential of the personality and the player.”
Aaronson's best position has been a point of contention, with the midfielder showing the work-rate and ability to play in a number of different spots.
The Union midfielder has primarily played as a number 10, but can also play as a number eight or in a wider role depending on the system.
Some have seen that versatility as a negative, with Aaronson yet to truly carve out a clearly-defined role due to his tendency to roam the field regardless of where he has been designated to play.
But Marsch sees it as an asset that can lead to him thriving during his time at Salzburg.
“I think he can play when we play with two 10s. I think he can play in that position,” Marsch said. "Sometimes we play with a number 10 underneath a striker or underneath two strikers. I certainly feel like he can play that way.
"He's played the number eight sometimes in a diamond with Philadelphia. I think he can do that really well. And I think he's capable of playing a six, especially if he has more defensive-minded six playing next to him and he's got a little bit more freedom to move around with and without the ball.
"So I think the combination of his natural abilities, his intelligence and again, his desire, his desire to be a big part of the game, his desire to have an effect in the game means that if you put him anywhere in the midfield, he's going to do well.”
Aaronson will finish out the season with the Union, who currently sit second in the Eastern Conference and are among the favorites to make a run in the playoffs.
"We want him to finish the year the right way. Certainly the relationship I have with Jim Curtin means that the last thing I'd want to do is put Jim's team in any kind of jeopardy by distracting one of their best players,” Marsch said.
"But it hasn't taken much at all. Brenden has been so focused. He's been so disciplined. He's given everything to the group there in Philadelphia. He plays every game and you can see how much heart he pours into every game.”