Kone was forced off with a slight thigh problem during his team's dramatic 3-2 win over Chelsea on Saturday, but Allardyce insists he is "okay for Wednesday", in a game where a win will preserve the club's Premier League status - at the expense of bitter rivals Newcastle United.
The Magpies were held to a disappointing 0-0 draw at bottom-of-the-table Aston Villa last weekend, meaning Sunderland require just two more points from their remaining two games in order to clinch survival.
Everton will arrive at the Stadium of Light having won just one of their last nine league games, but Allardyce has warned his players not to be complacent, despite handing them the day off on Sunday.
According to the Northern Echo, the former West Ham boss told his pre-match news conference: "I hope there is not a feeling out there that this game is a foregone conclusion.
"It will be anything but. Everton have been criticised for their performance against Leicester and I think that will get a reaction from their players and Roberto here on Wednesday.
"I think we will see Everton trying their best to spoil our evening. We must be in top form and be as committed as we were on Saturday, if we are to get the result we are looking for."
While Kone looks set to be involved on Wednesday, the same cannot be said for Billy Jones, who looks likely to miss out with a hip injury.