Twenty Twenty Media Vision, set up by MediaCom founder Bill Jones, has held the non-broadcast part of the business, which includes national and regional press, since March last year, when it took the brief from fellow Tunbridge Wells-based media agency WFCA.
Carpetright has already had preliminary conversations with several, unnamed media agencies, in advance of pitching the account.
Brilliant Media currently holds the TV and radio-buying account for the brand, which is understood to be part of a consolidated briefing to the agencies that have been approached.
Both incumbents expected to compete for the brief should it go to a full-blown pitch.
The retailer, which is set to unveil its full-year profits tomorrow (Tuesday), has issued three profit warnings over the past 12 months and is expected to announce 2010 profits just below the £17m it recorded in 2009.
The group was founded by Lord Harris of Peckham and has 560 outlets in UK and Ireland, and a further 120 in the Netherlands and Belgium.
It now has 230 shops selling the Sleepright brand, as the retailer attempts to move towards new revenue models.
A spokesperson for Carpetright said the approach to media agencies was part of a "market test".