Professor Steve Jones, of London University, says the idea is an "overstatement of what science, what biology can tell you".
"As most people know, chimpanzees share about 98% of our DNA, but bananas share about 50% of our DNA and we are not 98% chimpanzee or 50% banana, we are entirely human and unique in that respect," he said.
"It is simply a mistake to use an entirely human construct, which is rights, and apply it to an animal, which is not human. Rights come with responsibility and Ive never seen a chimp being fined for stealing a plate of bananas."
Archbishop of Pamplona and Tudela Fernando Sebastian said he could not believe it was even being proposed.
"We dont give rights to some people - such as unborn children, human embryos, and we are going to give them to apes," he said.