The Nationals said Monday that Madson, 37, was dealing with back issues, but the plan was for him to pitch through it. He is 2-5 with a 5.19 ERA and four saves in 48 games this season.
Glover, who missed most of the last two seasons because of shoulder and other assorted injuries, was promoted from Triple-A Syracuse on Wednesday, gave up a game-ending solo homer in the ninth inning of a 7-6 loss to the St. Louis Cardinals' 7-6 on Monday night.
Nationals manager Davey Martinez said Glover, who also pitched Sunday against the Cubs, is probably unavailable Tuesday after pitching back-to-back days for the first time this season.
Sean Doolittle, the Nationals' ace closer for the first half of the season, has been sidelined indefinitely because of a left toe injury. He hasn't pitched since July 7. Kelvin Herrera also is on the DL because of a right rotator cuff impingement.
The Nationals shuffled their bullpen mix Tuesday, recalling right-hander Trevor Gott and selecting the contract of left-hander Tim Collins from Triple-A Syracuse.
In addition to putting Madson on the DL, they also sent left-hander Sammy Solis back to Syracuse to open a second spot on the 25-man roster.
To create room for Collins on the 40-man roster, the Nats moved right-hander Erick Fedde from the 10-day disabled list to the 60-day disabled list.