A person familiar with the decision confirmed the choice Monday on condition of anonymity because a formal announcement had not been made. The school called a news conference for 2 p.m. ET Wednesday to introduce the new coach.
Williams will become only the second black head football coach in Ivy League history. Norries Wilson was the first when Columbia hired him in December 2005.
"I am happy for Tom," Jaguars coach Jack Del Rio said in a text message to The Associated Press. "He has shown strong leadership qualities and is a very good communicator. This is a great opportunity for him."
Williams replaces Jack Siedlecki, who retired in November after 12 seasons to become an assistant athletic director at Yale. Siedlecki was 70-47 at the school, including 47-37 in the Ivy League. He led the Bulldogs to a share of the conference title in 1999 and 2006, but was just 4-8 against rival Harvard and had lost seven of the past eight games against the Crimson.
Yale went 6-4 this season, including 4-3 in the conference and a 10-0 loss to Harvard in the 125th edition of "The Game."
Siedlecki became coach after Hall of Famer Carm Cozza retired in 1996 after 32 seasons.
Information from The Associated Press was used in this report.