The clash reportedly began when Rinder, acting as project manager, prioritised personal anecdotes about Christmas over strategic planning for their task.
The celebrity teams, featuring stars such as Jake Wood and Angela Scanlon, were tasked with creating gingerbread biscuits in Lapland.
Lord Sugar, alongside aides Brady and Mike Soutar, will oversee the specials, which mark the first time the public can purchase items made by the celebrities.
The two Celebrity Apprentice episodes are scheduled to air on BBC One on 29 and 30 December.