"He's arriving this afternoon in Moscow and we intend to meet tomorrow and we'll talk over all the details," Fursenko told reporters after easily beating his only rival Alisher Aminov for the RFU post, with 95 votes out of 106.
Russia failed to qualify for this year's World Cup in South Africa under Hiddink, whose deal expires at the end of June.
When Dutchman Hiddink arrived in Moscow he was warmly greeted by several dozen Russian fans, who shouted: "Only Guus, Only Guus, together we will win."
"It was a nice welcome," Hiddink told Reuters at the airport.
"I already know that Fursenko was elected the new boss of Russian football and we plan to meet on Thursday to discuss a number of things."
Asked about the reports, linking him with coaching jobs in Turkey and China, Hiddink said: "I've heard there is a lot of interest but I can't say anything until I speak with Fursenko.
"Yes, we (Russia) failed to qualify for South Africa, I also feel that I failed as a coach but now I feel refreshed and ready for a new challenge," he added.
"I feel like I'm not finished with my job here in Russia just yet and I intend to fulfil my contract at least until the summer.
"What happens after that we'll just have to wait and see."
Fursenko succeeded Sports Minister Vitaly Mutko, who resigned from his dual role as RFU president in November after Russian President Dmitry Medvedev demanded government officials in charge of national sports bodies be replaced by professional personnel.
The 55-year-old, a close friend of Russian Prime Minister Vladimir Putin, had been a clear favourite to replace Mutko after another leading candidate, Sergei Kapkov, an associate of Chelsea's billionaire owner Roman Abramovich, pulled out of the race.
Fursenko, an engineer who worked in the space industry, was a top manager at Russian energy giant Gazprom before being named Zenit president in 2005.
He held that job for three years, overseeing the club's most successful period.
In 2006, he hired Dutch coach Dick Advocaat, who the following year led Zenit to their first Russian league title in nearly a quarter of a century and then steered them to the UEFA Cup and the European Super Cup the year after.
