Dortmund have reportedly triggered a release clause worth around 22 million euros for the 27-year-old midfielder, bringing an end to his four-and-a-half-year spell in Eindhoven.
Veerman has admitted he was delighted to make the move and believes Dortmund represent the ideal next step in his career.
"Borussia Dortmund is a major club in Europe, and I'm proud to now be a part of it," Veerman said on Dortmund's official website. "I'm thrilled to be here and look forward to celebrating successes together with BVB."
Veerman joined PSV from Heerenveen in January 2022 and quickly established himself as one of the Eredivisie's leading midfielders.
His passing and creativity were among his biggest strengths, with the midfielder providing at least 10 assists in five of his last six Eredivisie seasons.
He leaves PSV having made 198 appearances, scoring 31 goals and providing 67 assists. He also helped the club win the Eredivisie title in each of the last three seasons.
Dortmund director Lars Ricken believes Veerman will strengthen a midfield that has undergone changes this year.
"Joey is a player whose skills will be of enormous help to us," Ricken said.
"We have deliberately focused our transfer activities this summer on central midfield, because with Pascal Gross and Salih Ozcan, we've lost two players there since the start of the year - and we plan to use Emre Can in central defence once he returns to the team."
Veerman had made no secret of his desire to play abroad, with previous interest from clubs including Brentford and Fenerbahce failing to result in a transfer.
The midfielder said the chance to join Dortmund was one he could not turn down.
"Sometimes an opportunity comes along that you simply have to seize. Borussia Dortmund is a perfect example of this," he said.
"For me, this is the next step I've always dreamed of, and I'm really looking forward to it. I'm going to do everything I can to make this adventure a success."
Veerman also used his farewell message to reflect on his time at PSV, where he won eight trophies and earned 16 Netherlands caps.
"PSV and Brabant have come to feel like home to my whole family and me. They were magical years, in every respect," he said.
The midfielder admitted leaving Eindhoven would be difficult after such a successful spell.
"At the same time, there are so many people at PSV whom I'm going to miss. I'd like to thank everyone for an unforgettable time. And I'm sure we'll see each other again very soon."
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