The 27-year-old has struggled to make the desired impact since joining the Toffees from Lokomotovi Moscow in a deal worth £13.5million in January 2016 and will be allowed to move on.
Niasse has still managed to chip in with six goals this season, including two against Bournemouth, and Everton will be looking for a fee in the region of £13million for the striker.
West Brom boss Alan Pardew is in need of goals to boost his squad and is said to be in the hunt, while Crystal Palace and Brighton are also keen.
Palace came close to signing Niasse in the summer, only for a deal to fall through at the last moment, but it is hoped the player will finally get a move away from Goodison Park in this window.
Everton could also look to move on Davy Klaassen and Sandro as Allardyce looks to trim his squad and recoup some cash in the process.