Allardyce took charge in December after Slaven Bilic was sacked, but Allardyce was unable to save the Midlands club from an immediate return to the Championship.
It is the first time the much-travelled Allardyce has been in charge of a team relegated from the top-flight.
"West Bromwich Albion made me a generous offer to stay at The Hawthorns that shows the commitment of an ambitious club," Allardyce said in a statement.
"After serious consideration I have decided not to accept that offer."
Allardyce, who has a reputation for guiding teams to safety, took charge with West Brom already in deep trouble, having won only one of their opening 13 games.
He improved performances, and in April, after back-to-back wins including an incredible 5-2 victory over Chelsea, it seemed that Allardyce might manage a great escape for West Brom.
But a 3-1 defeat by Arsenal on May 9 condemned them to a second relegation from the top-flight in four seasons.
"The club have asked for my opinion on the qualities my successor will require, and I have been more than happy to offer my thoughts," Allardyce said.
"The search for my replacement starts after the final game of the season at Leeds United.
"In the meantime, we will be concentrating fully on the preparation for that game and will do everything we can to finish the season with as many points as possible."
Allardyce's longtime assistant Sammy Lee and first team coach Robbie Stockdale will also depart the club.
Interviewed by Sky Sports after the defeat by his old club West Ham, an agitated Allardyce offered a rather ironic take on his decision to step aside.
"You all know I am a short-term manager, you have all seen me like that for the last six years," he said. "That is me, that is where I lie and that is what you continue to think.
"I haven't been more than that [a firefighter] for the last six or seven years because that is all you lot talk about."
He also took offence at West Ham striker Michail Antonio's description of West Brom's style as direct.
"If you thought that was long ball today then somebody is deluded," Allardyce said.
Sporting and technical director, Luke Dowling, said: "Sam, Sammy and Robbie came in at a really difficult time for the football club and worked tirelessly in their efforts to keep the club in the Premier League.
"Ultimately, we were unable to accumulate the points total required to keep us here, but Sam certainly restored pride with a number of resilient displays."