The incident took place after Fabio Carvalho's 98th minute winner
Member's of both teams' coaching staff clashed after Fabio Carvalho's 98th-minute winner.
The pair have been charged with improper behaviour.
Achterberg, Liverpool's goalkeeping coach, is also alleged to have used abusive and/or insulting gestures.
Hodges is Newcastle's head of performance.
Both have until 8 September to respond to the charge.
It follows a review by the FA into incident where a Newcastle staff member appeared to throw an object towards the Liverpool technical area.
Newcastle boss Eddie Howe called the late winner a "sickener" after his side had led 1-0 through a debut goal from £60m record signing Alexander Isak.
Roberto Firmino equalised for Liverpool before Carvalho hooked in from close range to win the game with almost the last kick.
-- Courtesy of BBC Sport