The deal for Musa, 23, will be announced at the weekend, for an undisclosed fee thought to be a Foxes record of around £16m.
BBC Sport understands he had offers from Southampton, Everton and West Ham.
Musa will be the club’s fourth summer signing, after Ron-Robert Zieler, Luis Hernandez and Nampalys Mendy.
CSKA Moscow coach Leonid Slutsky announced that the Nigerian, who can play as winger or striker, was leaving the club on Monday.
“If you look at the way things are done over there, you’ll see he is a player just made for English football,” Slutsky told the Russian media.
Musa joined CSKA in 2012 and scored 54 goals in 168 games, and has scored 11 times in 58 appearances for Nigeria since making his debut in 2010.
He scored twice in Nigeria’s 3-2 loss to Argentina at the 2014 World Cup in Brazil, becoming the first Nigerian to score twice in a World Cup finals match.
His first goal in that Group F game was his country’s fastest at the tournament, coming after just four minutes