They are submitting an application to the Intellectual Property Office in the UK regarding the matter.
The Telegraph suggests that the goal is to protect supporters from fake and third party merchandise.
The clubs may also want to make money based off of the Albion name.
Tottenham are an example of a club that was successful with recent trademark applications.
They trademark words associated with the team, including 'Tottenham', 'Tottenham Hotspur', 'Spurs', 'White Hart Lane', 'THFC', 'To Dare is to Do' and 'Hotspur'.
The three higher division Albions would not stop lower league Tadcaster Albion and Witton Albion from continuing to use the name.