The video mobile company, which trades under the brand 3, is involved in regulator Ofcom's wide-ranging review and consultation over its allocation of the radio spectrum.
An H3G spokesman said it was 'inviting credentials and expressions of interest' from agencies and that the PA review was in its 'early stages'.
Lexington Communications, which has held the account for more than three years since before 3's commercial UK launch, is being invited to retender.
Head of public policy Daniella Goodman, who joined H3G from Ofcom at the start of the year, is overseeing the review.
It comes as Gurling, head of public policy and government affairs, prepares to leave H3G later this summer after four years. It is understood he did not want to relocate from London to H3G's Maidenhead office. Gurling said he was 'talking to a number of people' over his next move and had not been made redundant.
He is understood to have held talks with eBay, which is seeking 12-month maternity cover for its head of PA Alison Scowen.
Prior to H3G, Gurling was an adviser to Liberal Democrat leader Charles Kennedy, now his brother-in-law (Gurling's sister is ex-Camelot PA chief Sarah Kennedy). He has also worked at GPC International and Hill & Knowlton.
H3G wants to ensure Ofcom's spectrum allocation minimises the potential for interference.
Parent firm Hutchison Whampoa has said it plans to float 3 next year in the UK.