"Markus is in the process of undergoing further medical tests and additional information will be made available at the appropriate time," the Cowboys said in a statement.
"The organization extends its prayers and support to the Paul family, and asks for friends and followers of Markus, his family and the team to keep them in their thoughts and prayers.
Dallas canceled Tuesday's practice earlier in the day but didn't provide details beyond saying the then-unnamed staffer's health issue wasn't related to COVID-19.
Paul joined Dallas as an assistant in 2018 and was named the strength and conditioning coordinator by first-year head coach Mike McCarthy during the offseason. He played as a defensive back for the Chicago Bears and Tampa Bay Buccaneers from 1989 to 1993.
The Cowboys are scheduled to play the Washington Football Team on Thanksgiving.