In return, the Nationals would receive several of the Dodgers' top prospects, sources told ESPN's Jeff Passan.
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Pitcher Josiah Gray, catcher Keibert Ruiz, pitcher Gerardo Carrillo and outfielder Donovan Casey are the four prospects the Dodgers will currently send to the Nationals, a source confirmed to Passan.
Scherzer is 8-4 with a 2.76 ERA in 19 starts this season. The three-time Cy Young Award winner is scheduled to become a free agent after 2021.
Scherzer's acquisition would give Los Angeles another ace starter in a rotation that is currently without Clayton Kershaw and Trevor Bauer.
Kershaw was placed on the IL July 7 with elbow inflammation, but threw three innings during a simulated game Tuesday - a big step toward his return. Bauer hasn't been with the Dodgers since initially being placed on administrative leave July 2 while sexual assault allegations made against him are investigated.
Turner, an All-Star this season, is batting .322 with 18 home runs and 49 RBIs. Now in his seventh season, the 28-year-old is a career .300 hitter. He exited Tuesday's game against the Philadelphia Phillies in the first inning following a positive test for the coronavirus.
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Turner would provide a boost to an infield that has been without Corey Seager for most of the season due to a fracture in his right hand.
The Dodgers currently have a 62-42 record heading into the weekend matchup with the Arizona Diamondbacks. Los Angeles is in second place in the N.L. West and trail the San Francisco Giants by three games.
ESPN contributed to this report.
The video in the media player above is from a previous report.