According to Patrick Mooney of The Athletic, the Cubs and Orioles are "deeply involved" in talks surrounding Britton.
The 30-year-old lefty is 1-0 with a 3.68 ERA, 1.36 WHIP and four saves in five opportunities this season.
Britton has appeared in just 15 games since an Achilles injury forced him to begin 2018 on the disabled list.
The two-time All-Star has shown in recent years that he is among the best relievers in baseball when healthy, however.
Between the 2016 and 2017 seasons, Britton successfully converted 60 straight save opportunities, which is an American League record.
In 2014 he had a 1.65 ERA with 37 saves, followed by a 1.92 ERA and 36 saves in 2015, and then a 0.54 ERA and 47 saves in 2016.
That 2016 performance was good enough to earn him fourth place in the AL Cy Young voting and win him the Mariano Rivera American League Reliever of the Year award.
He hasn't been quite as lights-out over the past two seasons, but injuries likely have plenty to do with that fact.
Last season, Britton was still solid with a 2.89 ERA and 15 saves in 38 appearances.
With current closer Brandon Morrow on the DL with biceps inflammation, the Cubs are in need of some help at the back end of their bullpen.
Veteran Steve Cishek has filled in admirably at closer, while Pedro Strop and Justin Wilson are enjoying solid years as well.
Britton is elite when he is at the top of his game, though, and adding him could give the Cubs a big leg up in the wide-open NL.
The Cubs currently have the best record in the National League at 57-39, and Britton would give them some late-game stability down the stretch and potentially heading into the playoffs.