Toffees boss Ronald Koeman has spent £90million so far this summer to bolster all areas of his squad as goalkeeper Jordan Pickford, defender Michael Keane, midfielder Davy Klaassen and striker Henry Onyekuru have been brought in, while Wayne Rooney returned on a free transfer from Manchester United.
However, with Romelu Lukaku departing in an initial £75million move to United, there is still scope to bolster the attack at Goodison Park.
Koeman is hoping to wrap up a swoop for Swansea playmaker Gylfi Sigurdsson, but wants another striker.
France international, Giroud, was believed to be a main target when he appeared to drop down the Arsenal pecking order after they brought in Alexandre Lacazette from Lyon.
But Gunners boss Arsene Wenger has stressed that he would be happy for Giroud to stay at the Emirates Stadium and the player would like to fight for his place.
Therefore, a move for Slimani has been mooted with the 29-year-old scoring seven goals and providing four assists in 23 appearances for Leicester after joining in a £28million deal from Sporting Lisbon last summer.
The Algeria international appears to be a big part of Foxes boss Craig Shakespeare's plans looking ahead to next season so it is likely that Koeman will have to pay over £30million to tempt their Premier League rivals into a sale.