Surrounded by family, Paul died Wednesday evening at Plano Presbyterian Hospital.
The cause of death was not announced.
"The loss of a family member is a tragedy, and Markus Paul was a loved and valued member of our family," Cowboys owner and general manager Jerry Jones said in a statement. "He was a pleasant and calming influence in our strength room and throughout The Star."
"His passion for his work and his enthusiasm for life earned him great respect and admiration from all our players and the entire organization. We offer our love and support to his family in this very difficult time. Our hearts are broken for his family and all of the individuals whose lives he touched and made better."
The Cowboys will recognize and remember Paul before their Thanksgiving Day matchup against the Washington Football Team at AT&T Stadium.
Paul was treated by Cowboys medical personnel and transported to a local hospital by ambulance shortly before 7:30 a.m. CT Tuesday. Coach Mike McCarthy canceled practice Tuesday, and Wednesday's session ran roughly 75 minutes, which was considered to be close to normal, given the circumstances.
Paul joined the Cowboys in 2018 as an assistant to Mike Woicik and was named the strength and conditioning coordinator upon McCarthy's arrival as coach.
"We extend our love, strength and support to Markus' family during this most challenging of times and ask that their privacy be respected moving forward," McCarthy said in a statement Wednesday. "Markus Paul was a leader in this building. He earned the players' respect and attention because he cared so much and was a naturally gifted communicator -- both on the personal and professional levels. He handled every situation, sometimes with a smile and a pat on the back and sometimes with tough love."
"He had innate toughness in a job that requires that quality, and he was admired throughout the NFL by his peers and the players he coached. It was a privilege to work with him as a coach and laugh with him as a friend. Markus did everything the right way."
Paul played five years in the NFL as a defensive back with the Chicago Bears and Tampa Bay Buccaneers, from 1989 to 1993, after four seasons at Syracuse, where he was a two-time All-American.
He entered the coaching ranks in 1998 with the New Orleans Saints and then spent five seasons with the New England Patriots' strength staff under Woicik from 2000 to 2004. In 2005 and '06, Paul was the director of physical development and head strength and conditioning coach for the New York Jets.
He spent 11 seasons with the New York Giants as an assistant strength coach before joining the Cowboys.
The Jets expressed condolences on their Twitter account Wednesday, describing Paul as "a kind man who made a lasting impact on those fortunate to have crossed his path."
Our thoughts and prayers are with the loved ones of Markus Paul, who spent two years with us in 2005-06. He was a kind man who made a lasting impact on those fortunate to have crossed his path.— New York Jets (@nyjets) November 26, 2020
The Giants also tweeted about Paul, calling him "a beloved member of our organization for several years" and saying, "He will be greatly missed."
In 2012, the Cowboys experienced the loss of a player one day before playing the Cincinnati Bengals, when practice-squad linebacker Jerry Brown was killed in a car accident in which teammate Josh Brent was the driver. The Cowboys had Brown's jersey displayed on their bench and won the game.