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CSKA take on the Londoners in Moscow on Tuesday in the first visit by an English side to the country since fans from the two countries clashed at the summer tournament in France.
"I believe that those English fans who come to Moscow to support their favourites will enjoy a high level of hospitality, not only at the stadium but everywhere," Slutsky, who managed Russia at Euro 2016, said at a press conference.
"I'm confident that they're in no danger. They have no reasons for fears."
But Slutsky hopes his players won't be quite so hospitable as they look to heap more misery on a Tottenham side that lost their opening Group E match at home to Monaco.
"Of course we should be very careful in dead ball situations," Slutsky said.
"The appearance of (Toby) Alderweireld, (Jan) Vertongen or (Vincent) Janssen in our penalty box will definitely be dangerous.
"But I don't think that the dead ball situations will be the only or main threat in tomorrow's match as Tottenham have a big group of skilled players in their attack and we need to shut them down."
Last week, CSKA crashed out of of the Russian Cup with a 2-1 away defeat to Yenisei Krasnoyarsk, which put an end to their eight-match unbeaten run in all tournaments.
The Russian champions also dropped points at the weekend after being held 1-1 by Krasnodar.
Slutsky said playmaker Alan Dzagoev, who missed Saturday's match with a persistent muscle problem, remains uncertain for the clash with Tottenham.
"We will finalise the decision on him in the morning (Tuesday)," Slutsky added.
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