Detroit sent the potential unrestricted free agent back to his original team in San Jose on Sunday in exchange for forward Andrew Murray and a conditional seventh-round draft pick in 2014.
Stuart had made clear that he wanted to be closer to his family, which still lives in San Jose, and now he gets three weeks to negotiate with the Sharks before possibly becoming a free agent July 1.
"That's the goal of making the trade," Stuart said. "We'll work on it for the next three weeks and try to work it out. We have some time to figure that out. For the Red Wings to give me time to figure that out shows what a class organization they are. I owe them a lot."
San Jose general manager Doug Wilson said he is hopeful the two sides can agree on a deal before the end of the month, which would give San Jose another top-four defenseman to upgrade a leaky penalty kill unit.
Stuart played more short-handed minutes than any other Red Wings defender this past season. San Jose had the second-worst penalty-kill unit this season, a major factor in their first-round exit from the playoffs against St. Louis.
"Brad is a player we are very familiar with -- a physical, team-first defenseman who is tough to play against, which is exactly the kind of mentality we want our team to possess," Wilson said.
Stuart said his priorities were finding a team that would be a title contender and one that was closer to his family. The Sharks fit in both categories, posting the second-best record in the NHL since the lockout.
"It's nice to know they're excited to have a chance to get me back," Stuart said. "I feel the same way. Hopefully we can work it out."
Stuart, 32, had six goals and 15 assists in 81 games last season. He had a plus-16 rating and led the Red Wings in hits with 177 and was second in blocked shots with 115.
He has played in 876 career games in 13 seasons with San Jose, Boston, Calgary, Los Angeles and Detroit, recording 74 goals, 231 assists and 489 penalty minutes. He's also added nine goals and 28 assists in 124 career playoff games.
"I take pride in being a good physical player who is hard to play against and works hard every night," Stuart said. "I want to be a guy the coaching staff and others can count on to give his all every night."