If St. Louis does not qualify for this year's playoffs, the Flames will receive St. Louis' fourth-round selection in this summer's draft, with the first-round pick being deferred to 2014.
The move comes after Calgary started to rebuild by trading captain Jarome Iginla to Pittsburgh last week.
The NHL trade deadline is 3 p.m. on Wednesday.
Bouwmeester, who has an annual cap hit of $6.68 million through to the end of next season, has six goals and nine assists in 33 games in 2013.
Bouwmeester joined Calgary in 2009 after being drafted third overall by the Florida Panthers in 2002. He has 71 goals and 229 assists in 10 seasons with the Flames and Panthers.