Woicik is scheduled to fly to Dallas Wednesday to talk to the Cowboys.
Woicik is the only player or coach with six Super Bowl rings, having won three with Dallas (1992, 1994, 1995) and three with the Patriots (2001, 2003, 2004).
He was the Cowboys' strength coach from 1990 to 1996 before spending the next three years in New Orleans. The Cowboys were 89-38 during his seven seasons, including 12 playoff wins.
A Massachusetts native, he joined the Patriots in 2000 when Bill Belichick took over as coach.
Woicik's contract with the Patriots was set to expire after the 2010 season, according to sources, and the sides have not reached an extension, opening the door for him to land elsewhere.
Woicik has a connection with Cowboys coach Jason Garrett, who joined the team as a backup quarterback in 1993.
Patriots spokesman Stacey James could not confirm Woicik's contract status.
Woicik, 54, was named the NFL's strength coach of the year in 1992 and 2004 by the Professional Football Strength and Conditioning Society.
Joe Juraszek has been the Cowboys' strength and conditioning coach since Woicik's departure, reuniting with Barry Switzer -- the two worked together at Oklahoma -- and continuing through the tenures of Chan Gailey, Dave Campo, Bill Parcells and Wade Phillips.
He missed most of the second half of the 2010 season with what was described as a "personal illness situation," according to the team. His assistant, Tony Ollison, took over the bulk of the work in Juraszek's absence.
Todd Archer covers the Cowboys for ESPNDallas.com. Mike Reiss covers the Patriots for ESPNBoston.com.