Jim Salisbury of NBC Sports Philadelphia reported the news, though he did not indicate the two sides were close on an agreement.
Britton, 30, is 1-0 with a 4.26 ERA and 1.34 WHIP in 13 appearances this season. He returned from a ruptured Achilles tendon in June.
Eleven of Britton's 13 appearances since his return have been scoreless. His blemishes were bad, including a miserable 0.1-inning outing against the Atlanta Braves in which he gave up five hits and four runs before being pulled. But Britton is rounding himself back into form with five straight scoreless outings and will be a target for relief-needy teams ahead of the July 31 deadline.
The Phillies, who are tied for first in the NL East at 51-40, need help in the back half of their bullpen. Philly ranks 17th in bullpen ERA and spent most of the early regular season struggling in its closer spot before settling on Seranthony Dominguez. Acquiring Britton would likely allow Dominguez to move back to an eighth-inning role, or the Phillies could approach the closer spot on a matchup basis.
Britton is a free agent after the 2018 season, so there is no long-term financial repercussions to a trade. The Orioles in all likelihood are only looking for a mid-level prospect or two to justify the move and might even be willing to package Britton and Manny Machado as part of a blockbuster.
If a contender wants to make a splash, Baltimore could be the first place to look.