The company is understood to be holding a pitch through matchmaker AAR for the UK consumer work.
Blue Rubicon, which PRWeek revealed last week had been put on alert by Giffgaff parent company O2 in a separate pitch, has worked on the account since 2009.
It has not repitched for Giffgaff, instead handing responsibility to sister agency Surname & Surname.
Surname & Surname co-founder Kev O’Sullivan explained this was because of his agency’s "focus on consumer".
News of the pitch comes as Giffgaff expands beyond its traditional heartland of SIM-only deals using the 02 network.
Last week the community-run mobile virtual network operator started selling a selection of 20 handsets.
Documents seen by PRWeek state that the company is looking to target the 18- to 24-year-old market in particular.
They point to Giffgaff’s "disruptive mindset", with "a shift in focus from strategic support to responsibility for all press office activities and brilliant execution of the strategy, plans and ideas generated in–house".
Blue Rubicon has worked on campaigns including The Chaos, which featured online video footage following a fictional band under the slogan ‘Life’s Better out of Contract’.AAR declined to comment, while Giffgaff had not responded to a request for comment at the time of publication.