The survey of 100 MPs was conducted by the Independent Parliamentary Standards Authority (Ipsa), the pay watchdog, with the politicians expressing their views anonymously. It also found more than one-third felt they should keep generous final salary pensions.
Conservatives wanted the biggest pay rise. They, on average, said their salary should be £96,740.
The Liberal Democrats thought the right amount was £78,361, while Labour opted for £77,322.
One MP suggested he and his colleagues should be paid £40,000 or less.
Some 5% said £60,000 to £65,000 was fair and 17% went for £65,000 to £70,000.
Just under 70% thought they were underpaid on the current salary of £65,738 with the average pay suggested being £86,250.
At present, the average wage in the UK is £23,000. Public sector workers have had pay frozen for two years and will see a pay rise cap of 1% for the next two.
Dave Prentis, Unison general secretary, said: "At a time when millions of workers are getting zero pay rises, the idea that MPs believe they deserve a 32% increase is living in cloud cuckoo land.
"MPs should get real about pay, this shows they are totally out of touch with working people. How can they think they deserve a 32% increase when the rest of the country is being told to tighten their belts?"
He added: "No wonder this research is anonymous; it shows real contempt for the plight of families across the country struggling to make ends meet."
Asked what David Cameron thought of the MPs' demand for more pay, the Prime Minister's spokesman said: "His view is that MPs' pay is a matter for Ipsa. The I in Ipsa stands for independent."