Sam Farmer of the Los Angeles Times reported the news and noted Kelley would not return.
Kelley emerged as the starter last season, and after neglecting to address the position in free agency or early in the draft, he entered the 2017 campaign in the same role.
So far this season, the former undrafted free agent out of Tulane ran for 30 yards in the opener against the Philadelphia Eagles and 78 yards Sunday before the setback.
As a rookie last season, Kelley registered 704 yards and six touchdowns on the ground, along with 12 receptions for 82 yards and one touchdown.
With Kelley potentially on the shelf, it opens the door for a veteran in Chris Thompson, as well as rookie fourth-round pick Samaje Perine.
Thompson is in his fifth season with the Redskins and scored two touchdowns on his first three carries in Sunday's game.
The 2016 season was the best of Thompson's career thus far as he set personal bests across the board with 356 rushing yards, 49 receptions for 349 yards and five touchdowns from scrimmage.
Thompson is Washington's third-down back due to his pass-catching ability, but Kelley's possible absence should get him far more opportunities on first and second down too.
Perine is the other running back who figures to factor into the equation since he is more of a prototypical runner than Thompson.
Perine was dominant in 2014 as a freshman with 1,713 yards and 21 touchdowns, but his production dropped precipitously over the next two years.
That resulted in his slipping in the draft, but the former Big 12 Offensive Freshman of the Year has undeniable physical ability that could shine through now that he is set to receive an opportunity.
Since the Redskins are a pass-first team, they figure to lean even more heavily on their aerial attack for as long as Kelley is unavailable.
That makes Thompson a highly valuable commodity, especially in point-per-reception leagues. If Thompson is available on the waiver wire, he should be added immediately.
Perine is worth a speculative add as well since he should get much of the short-yardage and red-zone work, but he should be deployed as a bench player initially, while Thompson should go straight into the lineup.