Bassett agreed to a three-year contract with Colorado through the 2024 season. Feyenoord has an option to purchase his rights, and the Rapids would get a percentage of future transfer fees.
A 20-year-old from Littleton, Colorado, Bassett made his MLS debut in 2018 and has 13 goals in 72 regular-season games.
He made his U.S. national team debut on Dec. 18 and scored in the 89th minute of a 1-0 exhibition win over Bosnia and Herzegovina.
Bassett was training with the U.S. in Phoenix and left on Jan. 12 for the Netherlands.
"This move also reflects our steadfast commitment to develop young players and provide a pathway to the highest levels of the game,'' Rapids general manager Padraig Smith said in a statement.