The Cowboys gave Moore a one-year deal, a source told ESPN NFL Insider Adam Schefter.
The team needs to clear a roster spot to make the move official. Owner and general manager Jerry Jones hinted at a deal for a veteran offensive lineman after Tuesday's practice but would not divulge a name.
Brian Waters' name has been linked to the Cowboys, but sources said there have not been meaningful discussions when his name came up last week.
Moore, 33, played for offensive coordinator Bill Callahan for four years with the New York Jets. He spent his first 11 seasons with the Jets, starting 142 of 144 games in his career and every game since 2005. Moore recently refuted a report that he would be retiring, telling ESPN New York that he was in shape and ready to go.
Nate Livings is expected to miss two to four weeks because of knee surgery last week, and Mackenzy Bernadeau returned to practice late last week after missing the first two-plus weeks with a hamstring injury.
Ronald Leary has worked with the first team in Livings' absence and has earned praise from the coaches. David Arkin started at right guard Sunday against Miami, but Bernadeau took all of the first-team snaps Tuesday.
The Moore signing would allow the Cowboys to keep rookie Travis Frederick almost exclusively at center. Frederick played some guard in the second half against the Dolphins, but the team wants to keep its first-round pick at center, if possible.
The Moore signing rekindles talk of last season's "Butt Fumble," a play in which Jets quarterback Mark Sanchez slid into Moore and saw the fumble returned for a New England touchdown.