David Hockney adored his dachshunds Stanley and Boodgie and let them run riot – so much so that after one had an accident in the LA home of the actor Dennis Hopper, the painter is said to have quipped: “It’ll be dry in the morning, luv, and you can just pick it up.”

But Hockney did not just indulge his four-legged companions, he treated them as portrait subjects for a series of paintings, one of which is to be sold by Christie’s.

The piece is the first from a series of dog paintings by Hockney, who died in June, ever to appear at auction. “This is such an exciting painting for us,” said Tessa Lord, Christie’s head of postwar and contemporary art. “The rarity is kind of extraordinary.”

The painting is titled Dog Painting 17 and shows Hockney’s first dachshund, Stanley, curled up on a cushion, fast asleep. It has an estimate of £1.5m-£2.5m and is one of more than 40 works that Hockney painted in 1994-95 for his Dog Series.

Stanley, named after Stan Laurel, was picked out as a puppy by Hockney in 1987 and followed him around constantly. Boodgie, more reserved, arrived later.

According to Hockney’s authorised biographer Christopher Simon Sykes, friends of the peripatetic Hockney were astonished he was getting a dog in the first place. But they soon saw how besotted he was.

The feeling was mutual and the noisy dachshunds soon became an essential part of his busy friend- and visitor-filled life in his Hollywood Hills home on Montcalm Avenue.

Sykes in his biography writes that Hockney was not bothered about house training, with one friend, Jonathon Brown, recalling how the dogs used to defecate in his shoes.

On a visit by Hockney for dinner at Hopper’s Frank Gehry-designed house in Venice, LA, Boodgie left a “gift” in the middle of the floor.

According to Brown, Hockney did retrospectively wag his finger, telling Boodgie “no, NO!”. Then, “as Hopper’s look of a chain-gang murderer increased in intensity, Hockney added: “Oah, it’ll be dry in the morning, luv, and you can just pick it up.”

Hopper, the story goes, was left with two choices: silence or murder. “He chose silence, but he looked like murder.”

Hockney embarked on his dog paintings when he needed some sunshine in his life. He had lost several friends to Aids and was devastated by the death of one of his closest, oldest friends, Henry Geldzahler, to pancreatic cancer. The artist told one journalist he wanted “desperately to paint something loving”.

Lord said: “He felt such a loss of love, this was kind of a catharsis for him.”

Christie’s believes there are only six of the dog paintings in private hands, with the others owned by the David Hockney Foundation.

The seller of Dog Painting 17 bought it after the dog series went on display at the LA Louver gallery, a wildly popular show that Hockney said attracted 7,000 people in four days – many with their own dogs in tow.

Lord said the painting displayed “incredible spontaneity” and “fluidity of brushstrokes, but it still feels richly textured. You’ve got this sort of mottling underneath the cushion and you’ve got that slight flick of the ear that really animates Stanley.”

She said the painting shone light on an important chapter in the life of one of the greatest artists of the 20th and 21st century. “I love the fact that there is this kind of innocence to it but it is so much more than that. There’s a universality to it … it’s so much more than just a portrait of a dog.”

The painting will be sold on 14 October during Frieze Week, and it will go on display – anyone can see it for free – at Christie’s in London from 8 October.