As Rosenthal notes, the Nationals are reluctant to add payroll mid-season. However, he also notes that Papelbon would make sense if the Phillies picked up most of his salary, which the Phillies are reportedly willing to do.
Papelbon, at leastaccording to Jim Salisbury of CSN, would not accept a trade unless he is promised to be the closer in his new destination. The Phillies have not approached him, Salisbury says.
The Nationals' current closer, Drew Storen, has done well in his job this season, so replacing him with Papelbon might not be the route to go. He's saved 29 of his 31 chances, while posting a 1.79 ERA (1.91 FIP) and 44 to 9 strikeout-to-walk ratio.
Papelbon has a 1.59 ERA (2.94 FIP) this season, saving all 17 of his chances this season. He's worked to a 40 to 8 strikeout-to-walk ratio in 39 2/3 innings.
If anything, the Nationals would likely use Papelbon as a setup man, which is not what he wants. In that case, it still would be interesting if he still accepted a trade if it came to that point.