The application, which will run to hundreds of pages, is expected to trigger a dispute between the local authority and central government over nuclear waste disposal.
Nirex's ultimate aim is to build a repository about 650 metres underground at the Sellafield site for the final disposal of Britain's intermediate-level radioactive wastes.
Some of the waste will be contaminated with plutonium and the repository will contain about seven tons of plutonium when full.
Nirex hopes to start operating the repository by about 2010.
Today's application for planning permission for the pounds 120m laboratory is an essential first step in the process. The laboratory, or 'Rock Characterisation Facility', will be used to carry out research on the rocks at depth and to trace the flow of underground water which might carry radioactivity back to the environment and contaminate human drinking water supplies.
Cumbria wants the Government to hold a wide-ranging public inquiry into how Nirex came to choose the Sellafield site and whether other areas in Britain might be safer in geological terms. It fears Nirex will become locked into the site even though others might be more suitable.