The leaflets promote the BNP ahead of the European Parliamentary elections on 4th June and carry an anti-immigration message. The Green Party and UK Independence Party are also using solus door drops to encourage people to vote for them in June.
Royal Mail has a legal obligation under the Representation of the People Act to deliver election material.
In asking Royal Mail post workers to deliver BNP leaflets, the Communication Workers Union is arging that Royal Mail is breaking a 'conscience clause' agreed several years ago allowing staff to refuse to deliver literature they find offensive.
The Times reported on Friday that some 100 workers in the West Country have told union leaders they will not carry the leaflets, but that the CWU is accusing Royal Mail chiefs in Bristol and Somerset of "bullying", with one office allegedly threatening workers with dismissal if they do not comply.
Dave Wilshire, spokesman for the Bristol and District Branch of the CWU, told the Times: "Over the past 48 hours I‚ve had around 32 calls from individual members and those speaking on behalf of groups, saying they don't want to deliver the material. That is fair enough.The clause says members don‚t have to deliver material if they feel threatened or if it is against their personal beliefs."
One Bristol based postman told the Western Daily Press that he was concerned about delivering inflammatory material as his route bordered two ethnic-minority areas.
The protest comes just after Tory leader David Cameron called for MPs' communications allowances, largely spent on door drops and direct mail, to be scrapped.