The all-rounder, 32, replaced Chris Gayle in 2010 and also led the team to the World T20 title in 2012.
"[The board is] looking to the future and I wish the new captain all the best," Sammy said in a video posted on his Facebook page.
West Indies play a two-match T20 series against India in Florida starting on 27 August.
"They've reviewed the captaincy of T20 and I won't be captain anymore, nor have my performances merited selection in the squad," he added.
"This is not me retiring from one-day or T20, it's just me thanking the fans and my players and the coaches I've worked with and West Indies Cricket Board (WICB) for letting me lead the side for the last six years."
Contacted by BBC Sport, the WICB said it could not confirm or deny Sammy's sacking.
In April, Sammy's side won their second World T20 by beating England in the final at Kolkata, with Carlos Brathwaite hitting the first four balls of the final over - bowled by Ben Stokes - for six.
Following the win, Sammy appeared to criticise the WICB, for which he was reprimanded by the International Cricket Council.
Darren Sammy says he has been sacked as West Indies' T20 captain four months after winning the World Twenty20.