Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
The former United captain, who made 707 appearances for the club, was replaced as assistant manager by Simon Donnelly when Jackie McNamara was brought in following the exit of Peter Houston. Darren Jackson also joined the coaching staff.
A club statement read: "With the installation of the new management team now complete, Dundee United confirm that an amicable agreement has been reached with former assistant manager Paul Hegarty.
"Paul, who also served Dundee United as player, club captain, coach and manager, leaves with the best wishes and grateful thanks for his many years of exemplary service on behalf of the club and will always be welcome at Tannadice Park."