The Falcons said in a statement that Smith was examined for a "non-emergency matter" but did not release details.
Atlanta general manager Thomas Dimitroff told Peter King of Sports Illustrated and NBC that Smith did not have a heart attack. Falcons spokesman Reggie Roberts would not confirm Smith was examined for chest discomfort.
Smith became ill and walked off the team plane after the Falcons' 31-23 win at Carolina on Sunday.
"Coach Smith is resting comfortably, his tests reaffirmed his health, and he is expected to be in Atlanta on Monday," the Falcons said in a statement released Sunday night.
Smith's status for the team's short work week before Thursday night's home game against Jacksonville was not known.
The 52-year-old Smith is in his fourth season with the Falcons.
Smith's staff includes two assistants with experience as a head coach. Offensive coordinator Mike Mularkey was Buffalo's coach from 2004-05. Receivers coach Terry Robiskie was an interim head coach with Washington in 2000 and again with Cleveland in 2004.
The Falcons had never achieved back-to-back winning seasons before hiring Smith from Jacksonville, where he was the defensive coordinator, in 2008.
With Smith on the sideline, the Falcons (8-5) are in position for their fourth straight winning season and second straight playoff berth. He was honored as NFL coach of the year by The Associated Press after leading the Falcons to an 11-5 record in 2008. The Falcons were 13-3 and had the top seed in the NFC last season before losing at home to Green Bay after a bye week.
The Falcons are two games behind New Orleans in the NFC South and in position for a wild card.