Costa Rica international Calvo, 26, played at last year's World Cup in Russia and was a 2018 MLS All-Star. The center-back scored five goals and set up four others in 60 regular-season matches for Minnesota since arriving at the end of 2016.
"We thank Francisco for his two-plus years of service with the club," United sporting director Manny Lagos said. "He was a big part of our growth into MLS and has been a great asset to our Minnesota community."
In 46 appearances for Costa Rica, Calvo has four goals, most recently in a friendly win last June over Northern Ireland.
Calvo made a name as a set-piece specialist with homeland club Saprissa, netting 14 goals over two seasons, nine of them in 2016.
"Acquiring Francisco's talents and character is a big plus," Fire president Nelson Rodriguez said. "He's a versatile defender with international experience in the prime of his career. He is familiar with MLS and has demonstrated a competitive mentality that should fit well with our group."
Minnesota has started the MLS campaign 4-3 with two drawn for sixth in the Western Conference on 14 points while the Fire are eighth in the Eastern Conference, just out of the playoff positions in the early going at 2-4 with three drawn.