The monkeys are snatched from the wild and trained to pick up to 1,000 coconuts a day, People for the Ethical Treatment of Animals (Peta) said.
The animal rights group said pigtailed macaques in Thailand were treated like "coconut-picking machines".
In response Waitrose, Ocado, Co-op and Boots vowed to stop selling some goods.
Meanwhile, Morrisons said it had already removed products made with monkey-picked coconuts from its shelves.
In a statement, Waitrose said: "As part of our animal welfare policy, we have committed to never knowingly sell any products sourced from monkey labour."
Co-op said: "As an ethical retailer, we do not permit the use of monkey labour to source ingredients for our products."
In a tweet earlier on Friday, the prime minister's fiancée Carrie Symonds, a conservationist, called on all supermarkets to boycott the products.
Sainsbury's subsequently told the BBC: "We are actively reviewing our ranges and investigating this complex issue with our suppliers."
Asda said: "We expect our suppliers to uphold the highest production standards at all times and we will not tolerate any forms of animal abuse in our supply chain." It pledged to remove certain brands from its shelves until it has investigated the allegations of cruelty.
Ms Symonds later took to Twitter again to urge Tesco to make a similar pledge: "Come on @Tesco! Over to you! Please stop selling these products too," she wrote.
A Tesco spokesperson told the BBC: "Our own-brand coconut milk and coconut water does not use monkey labour in its production and we don't sell any of the branded products identified by Peta.
"We don't tolerate these practices and would remove any product from sale that is known to have used monkey labour during its production."
1,000 coconuts a day
Peta said it had found eight farms in Thailand where monkeys were forced to pick coconuts for export around the world.
Male monkeys are able to pick up to 1,000 coconuts a day, Peta says. It's thought that a human can pick about 80.