Advocaat, 62, took over Belgium's national side in October 2009 and was under contract with the Red Devils until after the UEFA EURO 2012.
He has managed Belgium in conjunction with coaching Belgian club AZ Alkmaar, a situation which provoked a certain amount of criticism from some quarters.
Advocaat, who could be replaced by his current assistant and former Belgian international, Marc Wilmots, is well respected in Russia having led Zenith Saint Petersburg to the UEFA Cup title two years ago.
He is due to take over from his compatriot Hiddink, who is leaving to take charge of Turkey. As well as Zenith Advocaat has coached PSV Eindhoven, Rangers, Borussia Monchengladbach, and the Dutch national team.