A source told ESPN's Jerry Crasnick on Saturday that the Nationals are now having internal discussions to grant Papelbon's wish.
The 35-year-old Papelbon has a 2-4 record and career-worst 4.37 ERA with 19 saves in 37 games this season. His struggles before the trade deadline led to the team's decision to acquire All-Star closer Mark Melancon from the Pittsburgh Pirates.
In the two weeks since the trade, Papelbon has made just two appearances and Melancon has pitched in five games, allowing one hit over five scoreless innings while converting both of his save chances.
According to ESPN, the Nationals could release Papelbon as early as Saturday because the team will need to make a corresponding roster move when it recalls Reynaldo Lopez, who is scheduled to start Saturday night's game against the Atlanta Braves.
The controversial Papelbon, a six-time All-Star closer, was acquired in a trade with the Philadelphia Phillies on July 28, 2015. One week before the end of the 2015 season, Papelbon attempted to choke teammate and eventual National League MVP Bryce Harper during an altercation in the dugout when the two players got into an argument after Harper did not run out a pop fly. Papelbon was suspended seven games for his actions.
Papelbon, a 12-year veteran, is making $11 million in 2016, the final year of his contract. He has a career record of 41-36 with a 2.44 ERA and 368 saves with the Boston Red Sox, Phillies and Nationals.