In Chapter Two Lennie overhears Slim and Carlson talking about the litter of puppies Slim's dog had just delivered. Slim doesn't know what to do with the puppies. He tells Carlson his bitch had nine of them but he drowned four immediately because he didn't think Lulu could produce enough milk to feed so many. Lennie immediately looks to George with the unspoken plea to ask Slim to give him one of the puppies.
"Yeah!" George said, "I heard him, Lennie. I'll ask him."
George is probably thinking that having a puppy will keep Lennie out of trouble and that it is better for him to have a puppy than to keep playing with mice. Slim is probably glad to get rid of two mongrel puppies so easily. He can give one to Candy to take the place of the old dog Carlson is determined to shoot. He can give another to Lennie.
There is a much deeper reason in plot terms for Lennie getting one of puppies. Lulu keeps the litter out in the barn, and Lennie is constantly going there to play with the puppy that is now his. No one else has any particular reason to go to the barn, so it will make the perfect setting for the accidental murder that Steinbeck plans to have take place there. Curley's wife comes to the barn mainly because she is in the habit of hiding from Curley. She is also lonely, and when she finds Lennie there she takes advantage of the opportunity to talk with someone about her troubles. Unfortunately, she does not realize that Lennie can be dangerous.
This is exemplary of Steinbeck's naturalistic way of handling plot elements. There is nothing more natural than a female dog having a litter of puppies, and it is natural for the bitch to keep them in the barn where it is warm and there is a lot of soft hay. Steinbeck intended all along to have Lennie get into really serious trouble so that George would end up shooting him to save him from a lynch mob. Steinbeck wanted to have his novella end quickly and finally because he intended to convert the story immediately into a stage play, which was produced in New York in 1937, the same year the book was published. The book had to be short in order to be adaptable into a stage play lasting no more than about an hour and a half. This called for a "shotgun ending." So Lennie's getting the puppy was really intended to lead to Curley's wife's death and then to Lennie's death at the hands of his friend George.