Rapoport originally reported Jan. 11 that Williams was expected to get the position.
The Cleveland Browns fired him from his role as defensive coordinator following the 2018 season.
The Browns hired Williams as their defensive coordinator prior to the 2017 season, and he served as the interim head coach in 2018 after the firing of Hue Jackson. The Browns promoted interim offensive coordinator Freddie Kitchens to be the head coach.
Williams went 5-3 in eight games as head coach, which is a solid record with almost any team but especially notable in Cleveland considering the franchise won just one game in the previous two seasons combined.
It was also an improvement over his first stint as head coach when he went 17-31 in three seasons leading the Buffalo Bills from 2001 through 2003.
Williams has been a defensive coach for much of his career, serving as the defensive coordinator of the Tennessee Titans, Washington, Jacksonville Jaguars, New Orleans Saints, St. Louis/Los Angeles Rams and Browns since 1997.
The NFL notably suspended him in 2012 for his role in helping orchestrate the bounty scandal when he was with the Saints, in which players would be paid for hits that injured opponents. It was an indefinite suspension at the time, but the Rams hired him as their defensive coordinator heading into the 2014 season.
While he made headlines for the suspension, Williams guided the Browns to respectability in the 2018 season. The team has enjoyed just one winning season since 2002, but he helped install optimism while coaching Baker Mayfield and others in the second half of the 2018 campaign.
He has been either the head coach or defensive coordinator for seven different defenses that finished in the top 10 in the league in yards allowed, and he will look to bolster that side of the ball and help lead the Jets to the postseason in his new position.
New York hasn't been to the playoffs since the 2010 campaign and will need to take significant strides on the defensive side if it is going to compete in an AFC East that features Tom Brady and the New England Patriots as the dominant force.
The Jets were an ugly 25th in the league in yards allowed and 29th in points allowed during the 2018 season, and it will be up to Williams to turn those totals around as he returns to a familiar role of defensive coordinator.