Keith Pompey of the Philadelphia Inquirer was the first to report New York's intention to cut ties with Sessions.
Yahoo Sports' Shams Charania added that the Knicks and Burke have agreed to a deal for the remainder of the 2017-18 season.
Burke previously played for the Westchester Knicks, the New York Knicks' NBA G League affiliate.
The 31-year-old Sessions started the first three games of the season for New York before falling out of the rotation.
He has appeared in 13 games this campaign, averaging just 3.7 points, 2.1 assists and 1.4 rebounds, all of which are career lows.
Burke is a 25-year-old University of Michigan product whom the Minnesota Timberwolves selected with the No. 9 overall pick in the 2013 NBA draft.
Burke signed with the Knicks in October and was assigned to the G League. In 26 games this season, he is averaging 26.6 points, 5.4 assists and 3.3 rebounds per contest.
Burke entered the Association with plenty of fanfare after being named the National College Player of the Year, and he enjoyed a solid rookie campaign with 12.8 points, 5.7 assists and 3.0 rebounds per game.
During his NBA career, he has put up 10.6 points, 3.6 assists and 2.1 rebounds in 267 contests.
Burke will likely serve in a backup capacity to veteran Jarrett Jack and rookie first-round pick Frank Ntilikina.