Pulisic is reportedly being tracked by a number of European clubs, after a whirlwind two-and-a-half years in the professional game.
The 19-year-old has scored 12 goals and provided 20 assists in 97 appearances in all competitions since making his Bundesliga debut for Dortmund in January 2016, and is a 21-time USA international.
For Arena, who coached Pulisic during his second spell as U.S. coach between November 2016 and October 2017, any move away from BVB could hinder the teenager's progress.
"I think Christian is in a great place," Arena told Omnisport.
"Dortmund has a very good reputation for developing players. In his couple [of] years there with the first team he's done very well. Hopefully he can continue to do that, be consistent at club level and if he does that he's going to be a real plus for the national team program.
"There's no point in him leaving Dortmund and going to a club where he doesn't get the kind of minutes he needs to continue to develop."
Watch: Pulisic played for Dortmund on American soil for the first time in May
Arena's words echo those of Pulisic's father, who earlier this month labelled rumours linking his son with a summer transfer to the English Premier League as "hogwash".
"He's linked with a different club every week," Mark Pulisic said.
"Right now, Christian plays for Dortmund and that's where he is planning on playing again next year."
With the USA having missed out on qualifying for the 2018 FIFA World Cup in Russia, Pulisic is currently enjoying a well-earned break ahead of the 2018/19 Bundesliga campaign - what will be his third full season of top-flight football.
The Dortmund star is being tipped to inspire a new generation of U.S. players to the 2022 World Cup in Qatar before taking centre stage at the 2026 edition, which the Stars and Stripes will co-host with neighbouring Canada and Mexico.