The Kings received permission from the Clippers to speak with Woodson, Yahoo Sports reported.
The Kings are seeking a replacement for George Karl, who was fired after the season.
The instability of the Kings is reflected in the revolving door of coaches. The new hire will be the ninth in the past 11 seasons.
Woodson, 58, led the Atlanta Hawks to three playoff appearances as their coach from 2004 to 2011. His overall record with the Hawks was 206-286.
Woodson led the Knicks to 54 wins the following season and to the second round of the playoffs, but in 2013-14 the team finished with a 37-54 record and his time with the Knicks ended with a 109-79 record.
For the past two seasons, Woodson has worked on Doc Rivers' Clippers staff as an assistant.
A job in Sacramento would be a homecoming of sorts for Woodson, who played for the Kings from 1981 to 1986.