More than half of the 153 MPs surveyed were also in support of Ofcom assuming a regulatory role.
The study was carried out by Communicate Research.
'There is very clear political support for dismantling the NLA's monopoly. There is also a clear appetite for stripping the NLA of its pseudo-regulatory functions and handing them to Ofcom,' said CIPR head of public affairs Francis Ingham.
Ofcom comms director Matt Peacock said: 'It would be for parliament to decide, but we have made it clear previously that we have no desire to regulate the newspaper industry – we cover electronic communications.'
The NLA claims that for the 'collecting society' model to exist, it must operate as a monopoly. If not, each cuttings agency would have to
contact each publication it monitors for a licence.
The CIPR is lobbying to have the NLA scrapped following a row over the overhaul of the NLA's service. It has digitised its system with the eClips programme, prompting concerns over price hikes, lack of confidentiality and censorship.
However, the NLA, which launched eClips this week, claims 40 per cent of the press cuttings industry has signed up to the system.