Russian media said Zenit St Petersburg midfielder Artur Yusupov is likely to be called up to replace Denisov in the 23-man squad with an official announcement expected on Monday.
Russia start their campaign against England on June 11 in Marseille and the Serbia game was their final warm-up.
Russia's Artem Dzyuba scored in the 85th minute at the Stade Louis II in Monaco but Alexander Mitrovic equalised three minutes later for Serbia, who failed to qualify for the Euros.
Slutsky kept faith with the attacking trio of Dzyuba, Fedor Smolov and Alexander Kokorin, who all started the 2-1 defeat by the Czech Republic last Wednesday.
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Igor Akinfeev returned in goal, while 20-year-old Alexander Golovin was also given a chance in the centre of midfield.
There were few chances for either side in what was a largely forgettable first half before Russia were hit by a major injury blow when 32-year-old Denisov, who plays for Dynamo Moscow, limped off during the second period.
Dzyuba scored after excellent work by Roman Shirokov and Kokorin but they three minutes later Mitrovic sprung the offside trap and found himself one-on-one with Akinfeev. The Newcastle United forward made no mistake as he scored to make it 1-1.
Russia and England are in Group B at Euro 2016 alongside Slovakia and Wales.
(Reporting by Dmitriy Rogovitskiy; Editing by Clare Lovell and Ken Ferris)
