The 39-year-old Russian played the past three seasons for Ottawa, recording 15 goals and 76 assists.
Gonchar, whose veteran presence on the Senators' blue-line helped offset the loss of Erik Karlsson for much of the regular season, is seeking a two-year deal. He's set to be an unrestricted free agent July 5.
"We thank Sergei for his efforts in Ottawa," Senators general manager Bryan Murray said in a statement. "We felt that this trade will give Sergei the opportunity to extend his career in the NHL and we wish him success with the Stars."
Reports indicated he was considering signing in Russia's Kontinental Hockey League.
Gonchar completed the final season of a three-year, $16.5-million contract. His cap hit was $5.5 million.
If Gonchar signs with Dallas, it will be his fifth NHL team. The sixth-round pick is conditional on Gonchar signing a contract with the Stars.
Gonchar was originally selected 14th in the 1992 draft by the Washington Capitals. He won the Stanley Cup with the Pittsburgh Penguins in 2009.
In 1,177 games over 18 NHL seasons, Gonchar has recorded 217 goals and 558 assists.