The Publicity Bureau in Doncaster, the incumbent, will repitch for the assignment against two Manchester shops - Cheetham Bell JWT and BDH\TBWA.
Senior managers from the competing agencies are meeting executives of the 150-strong chain over the next few days to be briefed.
Computer World was established after Time Computers' acquisition of its rival, Tiny, at the beginning of last year. The outlets house both companies' products.
However, ComputerWorld has found it hard to match the marketing resource of the Dixons-owned PC World, which spends more than £25 million a year through M&C Saatchi and Walker Media.
Industry sources claim that Computer World has suffered as a result of a marketing strategy that is confused and the difficulty of getting to grips with retail because of its background in direct selling.
"PC World hasn't only been very successful with its marketing, but very canny with its pricing strategy," an insider said. "Computer World just doesn't spend enough to compete."
Some also believe the store's offering is less attractive than that of PC World, where customers can choose their computers and carry them away from the store. Computer World customers have to order what they want, then wait for them to be delivered to their homes.
Brian Trevaskiss, Computer World's marketing manager, was unavailable for comment as Campaign went to press.