The news comes after the Sabres lost their eighth straight game (0-7-1) 5-4 in a shootout to the Philadelphia Flyers.
Eichel appeared to hurt his neck after being checked into the end boards by Casey Cizikas with about four minutes left in a 5-2 loss to the New York Islanders on Sunday.
The 24-year-old captain has already lost two games to a lower-body injury and the first week of training camp to an upper-body injury.
Eichel, an elite goal-scorer, has struggled to find the back of the net this season, potting just two goals in 21 games, and is mired in a 13-game scoring drought. Despite that, his 18 points lead the team.
The second pick in the 2015 NHL Draft put up a career-best 1.15 points per game in the pandemic-shortened 2019-20 campaign, as he racked up 36 goals and 78 points in 68 games.