The pitch is being fought by regional specialists Beattie Communications, Golley Slater and Harrison Cowley, plus the Leeds-based incumbent.
The value of the account is believed to be in the region of £350,000, and is thought to involve both PR and broader marcoms support.
The pitch is being overseen by head of marcoms Melanie Russell, and a decision is expected within three weeks. Anglian declined to provide details of the brief.
The account is a significant one for Brahm, which has links with 40-year-old Anglian stretching back to the late 1990s when it was brought in to handle creative work for the company. It still handles advertising and design work, as well as Anglian's PR.
The firm is the UK's largest maker and supplier of replacement doors, windows, conservatories and roof-trims. But the reputation of the industry in which Anglian operates suffers from pressure-selling and complex deals.
Previous Brahm-handled campaigns have sought to position Anglian as ‘the householder's friend'.
Anglian unveiled sponsorship deals with two Scottish football clubs in August (Motherwell and Dundee United) and is thought to be ready to invest heavily in marcoms.
Russell is the firm's top communicator following the departure of marketing director Andrew Pound in June. However, it has just signed up Martin Troughton to fill the role. He is set to join in November.
The company got a new CEO in February, when ADT Fire and Security Systems president Alex McNutt replaced Larry Condon.