The 64-year-old has been GM since the 2003-04 season. The Sharks reached Stanley Cup Playoffs in 14 of his 17 seasons but has not qualified the past two seasons.
San Jose assistant GM Joe Will is taking over day-to-day operations.
"We're just thinking about him. We want him to get better," Sharks coach Bob Boughner said following a 4-1 loss to the Toronto Maple Leafs on Friday. "The most important thing is health. In this game, there's not a lot of time to rest and try to get better. It's good he's taking some time away, and we can't wait to see him back."
The Sharks are 741-445-155 with Wilson as GM. The wins are No. 16 among GMs in NHL history.
Wilson scored 827 points (237 goals, 590 assists) in 1,024 regular-season games during 17 NHL seasons as a defenseman with the Chicago Blackhawks and the Sharks from 1977-93. He was elected to the Hockey Hall of Fame in 2020, with the induction ceremony taking place this month after being delayed by the coronavirus pandemic.
San Jose (10-9-1) is currently fifth in the Pacific Division.
"Just want to make a quick statement on behalf of our team that we respect his privacy," Sharks captain Logan Couture said. "We love him, we're going to miss him, every guy in that room owes a lot to him. We hope he gets better and we see him soon."
