The majority of the industry is regulated by the Independent Press Standards Organisation (Ipso), which is funded by its members.
Rival regulator Impress currently has a small number of, mainly hyperlocal, members.
Major newspapers have refused to sign up to Impress because they see it as a step towards state regulation and take issue with the role of Max Mosley, who partly funds it.
A YouGov poll, commissioned by the News Media Association, has found public opposition to this structure.
Some 49 per cent of 1,632 UK adults thought the industry should fund the regulator, ahead of 25 per cent who said general taxation, and four per cent who said “donations from wealthy individuals and trusts”.
Lynne Anderson, News Media Association deputy chief executive, said:
This survey demonstrates conclusively that a regulatory regime led by Impress – which is completely reliant upon funding from one wealthy individual, Max Mosley – cannot command the confidence of the public.
Ipso is funded in its entirety by its member national, regional and local newspaper publishers which is the funding model the public want and expect from an industry which is committed to robust self regulation.
The poll also found that press regulation is a low-priority area for the public. One per cent of those polled said it should be a top-four priority area for the government over the next few years, behind the likes of Brexit (53 per cent), crime (17 per cent), animal rights (five per cent) and airport expansion (two per cent).
Those surveyed were also asked about Facebook news story regulation. Some 56 per cent said news on Facebook should have the same level of regulation as newspaper stories.