"I fully expect him to be in our lineup come opening night next fall," BriseBois said in comments published Wednesday.
The deadline to ask a player to waive his no-move clause for the Seattle expansion draft was Tuesday at 5 p.m. ET.
Stamkos, a 31-year-old forward, has three seasons remaining on the eight-year, $68 million contract ($8.5 million average annual value) he signed with Tampa Bay on June 29, 2016.
During the expansion draft, which will be held July 21 (8 p.m. ET; ESPN2, SN, SN NOW), the Kraken will select one player from each team excluding the Vegas Golden Knights for a total of 30 (at least 14 forwards, nine defensemen and three goalies).
The other 30 teams can protect seven forwards, three defensemen and one goalie, or eight skaters (forwards/defensemen) and one goalie. The deadline for teams to submit protected lists is Saturday at 5 p.m. ET.
"Seattle is in the driver's seat. They get to barter with 31 teams and shop one team against another," BriseBois said. "And a lot of teams are probably trying to do similar things as we are. I think every team would like to work out a deal with Seattle where it's win-win.
"But it's probably going to be impossible for 30 deals to be cut here. I'd like to cut a deal with Seattle, but we may not be able to. At the end of the day, the way it's set up process-wise is that we'll submit a list, and it's very possible that Seattle will just claim a player, and we're going to move on with the rest of our reserve list.''
Stamkos scored 34 points (17 goals, 17 assists) in 38 games this season, missing significant time while on long-term injured reserve with a lower-body injury. He returned for the playoffs, though, and scored 18 points (eight goals, 10 assists) in 23 games to help the Lightning win their second straight Stanley Cup championship.
Selected by Tampa Bay with the No. 1 pick in the 2008 NHL Draft, Stamkos has scored 866 points (439 goals, 427 assists) in 841 regular-season games and 72 points (32 goals, 40 assists) in 94 playoff games. He has been their captain since March 6, 2014.
On Tuesday, BriseBois said that the Lightning will have to "get creative" for next season to be under the NHL salary cap, which will remain at $81.5 million.
"We have a Stanley Cup-winning roster, and our challenge to maintaining that roster is the salary cap," BriseBois said.
Forward Tyler Johnson, who turns 31 on July 29, has three seasons remaining on his contract with a $5 million AAV. The Spokane, Washington native is considered to be a possible selection by Seattle.
"In fairness, Tyler was on the fourth line once [Nikita Kucherov] came back (for the Stanley Cup Playoffs)," BriseBois told The Athletic. "He played in our top nine for most of the season. And in the previous year, he was on our second line when we won the Stanley Cup. The reality is, Tyler is in the right chair when he's playing center. Once we had our full roster come playoff time, the best place for him to help our team win was playing in the middle on that fourth line."
