The former Celtic youth team player has spent the last seven years coaching in the USA and is currently in charge of Miami FC.
Livingston sacked Kenny Miller on Sunday after seven weeks in charge of the newly promoted club.
Another former Celt of the mid-nineties, Gary Holt, has been strongly linked with what must be one of the toughest jobs in Scotland.
Livingston have survived a number of financial problems over the last 15 years and bounced into the Premiership on the back of successive promotions.
David Hopkin quit soon after defeating Partick Thistle in the Premiership play-off with the West Lothian side now looking for their third manager of the year.
After playing for Celtic in the 95/96 season Paul moved on to play for Newcastle and Norwich before returning to Scotland where he had stints at Kilmarnock, Livingston and Hibs.