Couture was named the 10th captain in Sharks history on Sept. 12, 2019, replacing Joe Pavelski, who signed a three-year contract with the Dallas Stars on July 1 of that year.
Thornton signed a one-year, $700,000 contract with the Toronto Maple Leafs on Oct. 15. The 41-year-old scored 31 points (seven goals, 24 assists) in 70 games with the Sharks last season, his 15th with them.
Thornton was Sharks captain from 2010-14 until he was stripped of the title in August 2014. He was one of four alternate captains in 2014-15, when the Sharks did not name a captain, and kept that title until 2020.
The Sharks have not named an alternate captain to replace Thornton.
"We've talked about that as a staff already and we wanted to see how first week or two play out in training camp," Boughner said. "We have some options we can go with, we don't have to name one, we can name one, so I think we're just sort of leaving that as something we'll tackle here in the next week or two."
Selected by the Boston Bruins with the No. 1 pick of the 1997 NHL Draft, Thornton has scored 1,509 points (420 goals, 1,089 assists) in 1,636 games during 22 seasons with the Sharks and Bruins. He has scored 133 points (31 goals, 102 assists) in 179 NHL playoff games.
Acquired in a trade from the Bruins on Nov. 30, 2005, Thornton is the Sharks leader in assists (804), second in games (1,104) and points (1,055), and fourth in goals (251).
"He's definitely left his mark here and Logan and Tommy and guys who grew up under him understand what it is to be a leader, a guy who comes to work every day, who pushes his teammates to be better," Boughner said of Thornton.
"Everyone has his own style. Couture may not have as many words as Jumbo, but when he gets on the ice he leads by example and he says what needs to be said in the room. There has to be a support system underneath Logan and that's going to come from the coaching staff and also the players around him, and they have a lot of respect for him. I think what's going to happen is that, last year wasn't a perfect year for us and it's my job and my staff's job to make sure Logan has the full support, he has the environment to be a good leader, and the rest of his leadership group."
Couture missed 17 games from Jan. 9-Feb. 23 due to a fractured ankle. He was also injured when a deflected puck hit him in the face against the Colorado Avalanche on March 8, San Jose's final game before the season was paused due to concerns surrounding the coronavirus. Hertl missed the final 18 games of the season after having surgery to repair a torn ACL in his left knee sustained against the Vancouver Canucks on Jan. 29.
Karlsson missed San Jose's last 13 games of the season after breaking his thumb against the Winnipeg Jets on Feb. 14; the injury required surgery. He was limited to 53 games in 2018-19 because of a groin injury that required surgery after the season.
