Slutskiy announced he would not extend his contract after Russia were knocked out of the European Championship last week, finishing bottom of Group B with one point from three games.
“The RFU would like to thank Leonid Slutskiy, who helped Russia to qualify for the 2016 European Championships, for his professionalism and we wish him all the success in his coaching career,” the RFU said in a statement on its website (www.rfs.ru).
“Information regarding the new coaching staff of the Russian team will be published on the RFU’s official website as soon as there is any information,” it added.
The 45-year-old, who is also CSKA Moscow coach, had been in charge of the national team since August 2015.
His resignation follows England manager Roy Hodgson opting not to seek reappointment after his side sensationally crashed out against minnows Iceland this week.
England and Russia collectively made more headlines for their fans’ loutish behaviour than for any success on the field during the tournament.
And Spain’s coach Vicente del Bosque is stepping down from the job just three days after the defending champions were knocked out of Euro 2016 by Italy in France.
“Of course, I have no intention of continuing,” he said in an interview with Radio Nacional de Espana.
“It was a decision I had taken beforehand.”
Spain, who were trying to win their third straight European title, lost 2-0 to Italy in the last 16 on Monday.
Del Bosque, 65, led Spain to their first World Cup title in 2010 and followed that success by clinching Spain’s second successive European Championship two years later.
At that point, Spain were the dominant side in world football, with their high-tempo possession game providing a template that many teams tried to emulate.
Their air of invincibility was punctured at the World Cup in 2014, when they were knocked out in the group stage after losing their opening two matches.
Del Bosque, who won two Champions League crowns and two La Liga titles in a four-year spell in charge of Real Madrid from 1999, was persuaded to stay on as Spain boss and led them into Euro 2016 where they reached the last 16.
They were well-beaten, however, by a well-organised Italy side and looked a shadow of Del Bosque’s previous teams.
“However the European Championship worked out, I had no doubt about my future,” he said.
“It was a decision I had taken beforehand. I don’t want to hide, nor make an exhibition of myself.”
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