The website reported Boucher has resigned from his post with the Montreal Canadiens' American Hockey League affiliate and will be introduced in Tampa at a news conference Wednesday.
Boucher, who just completed his first season as a head coach in the pro ranks, was offered the Columbus job last week and asked the Blue Jackets if he could think about it over the weekend.
Instead, Boucher will become the first head coach under new general manager Steve Yzerman. Yzerman was hired last month after spending five years in Detroit as vice president for the Red Wings.
The Lightning were 12th in the Eastern Conference last season and new team owner Jeff Vinik fired head coach Rick Tochett and general manager Brian Lawton after the season.
Tampa Bay has made the playoffs twice since winning the Stanley Cup in 2004 but has not won a playoff series.
Boucher takes over a roster including two stars from the Stanley cup winner, Vincent Lecavalier and Martin St. Louis, as well as Victor Hedman and Steven Stamkos, the top pick in the 2008 NHL draft who scored 51 goals this season to tie Sidney Crosby for the league lead.
Information from The Associated Press contributed to this report.