Sources told ESPN that Club America will pay $6 million, with LAFC also receiving a percentage of any future sale.
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"Brian has been an important piece of our attack and a huge reason for all of our success since his arrival in 2019," LAFC president & general manager John Thorrington said in a team release. "He is a player who is capable of special things, and this move will be a great opportunity for him to continue to develop in a competitive environment."
The 22-year-old Uruguayan winger arrived at LAFC in August of 2019 with big expectations after the MLS side spent around $8 million to secure him from Penarol and make him a young designated player.
Rodriguez, a three-time member of the league's 22-Under-22 list, helped push his team to the Supporters' Shield in his first season. In 2021, he had a brief stint on loan with Almeria in Spain's second division.
Rodriguez told TUDN earlier this month that he would be open to a move to Club America, noting that it "would be a great leap."
The winger will look to regain the attention of Uruguay's national team after being left out of World Cup qualifiers and friendlies in 2022.
Rodriguez first debuted with Uruguay in 2019, but has failed to earn international minutes since late last year. Later this year, Uruguay will kick-off their World Cup run with group stage matches against South Korea, Portugal and Ghana.
National team manager Diego Alonso previously coached in Liga MX with Pachuca and Monterrey, as well as with Inter Miami in MLS. At Club America, Rodriguez will join fellow Uruguayans Jonathan Rodriguez, Sebastian Caceres and Federico Vinas.
The giants from Mexico City currently sit third in the 2022 Apertura standings after winning its last five games
LAFC, along with Rodriguez, recently met Club America in a friendly on Aug. 3 at Los Angeles' SoFi Stadium. In a double-header friendly that began with LA Galaxy defeating Chivas 2-0, Club America narrowly beat LAFC on penalties after a 0-0 draw in regulation time.
Information from ESPN's U.S. soccer correspondent Jeff Carlisle was used in this report.