"It's true that I will leave my job after the European championship," the 64-year-old, who had replaced fellow Dutchman Guus Hiddink following Russia's failure to qualify for the 2010 World Cup, told local media.
"I don't want to give the reason just yet but one thing I can say for sure - it's not because of money."
Advocaat said he had agreed to join his former club PSV, whom he led to the Dutch title in 1997, following June's Euros.
"I have reached a verbal agreement to coach PSV, all we need now is to sign a formal contract," he was quoted as saying by Dutch media.
Advocaat has also had spells as manager of Netherlands, the UAE and South Korea as well as successful stints at club level with the likes of Glasgow Rangers as well as PSV.
Russian FA chief Sergei Fursenko, who worked with Advocaat at Zenit St Petersburg, persuaded the Dutchman to quit his job as Belgium coach after just six months by offering him a two-year contract with an option to extend it for another two years.
In 2006, then Zenit president Fursenko hired Advocaat and in his first full season he steered the club to their first national title in nearly a quarter of a century, becoming the first foreign coach to achieve the feat.
In 2008, he led the side to victory in the UEFA Cup.
Local media reported that the relationship between the pair had cooled in recent months, prompting Advocaat to leave on his own rather than being forced to do so.
Reports said Advocaat wanted to sign a new deal before the Euros while Fursenko preferred to wait and would only offer a new contract if Russia reached the quarter-finals in Poland and Ukraine after reaching the last four at Euro 2008 under Hiddink.
"During the negotiations with Dick Advocaat (the Russian FA) had offered to prolong his contract after Euro 2012," Fursenko told the FA's website.
"At the same time, the coach had a right to make his own decision before the start of the Euros. Dick chose to exercise his right. We respect his decision."
Reports also said Fursenko would likely to hire a Russian coach this time, with Russia under-23 boss Nikolai Pisarev being mentioned as a front-runner for the job.