Passan reports that the deal is pending medical reviews.
Alex Verdugo will be the centerpiece of the trade coming back to Boston.
The details of the trade have not been announced.
With the bidding for Betts heating up between the Dodgers and the National League West rival San Diego Padres in recent days, Boston chose to deal with Los Angeles and improve the team that lost in the division series to the Washington Nationals, the eventual World Series champions. Right-hander Kenta Maeda would move from the Dodgers to the Twins, with right-hander Brusdar Graterol moving from Minnesota to Boston.
Betts, a free agent after the season, repeatedly turned down contract extension offers from the Red Sox and is expected to command a massive contract in the offseason.
Still, Boston is one of the wealthiest franchises in baseball, and their decision to trade their homegrown, 27-year-old former MVP to avoid paying the luxury tax is not sitting well with a lot of baseball fans, who they think should have just paid the man what he deserves. Just in the a year ago, Los Angeles has added Jalen Ramsey, Kawhi Leonard, Paul George and Anthony Davis - and now Betts.
Verdugo didn't play after August 4 because of a back injury he re-aggravated while on a rehab assignment in September. Price is no longer the Cy Young candidate of his earlier years, but he has posted a 3.84 ERA in four seasons since signing with Boston ahead of the 2016 campaign.
With the Red Sox clearly in at least a partial rebuilding phase, their World Series odds have taken a substantial hit.