This is a good question. There are several places in the book where Steinbeck shows that George might have to do something with Lennie.
First, right from the beginning, George and Lennie are on the run because Lennie got in trouble with a young woman in their previous place of employment. Lennie has a habit of touching soft things. So, when he touches a woman's dress, she becomes afraid. The people think that he is going to rape her. So they run away.
Here is what George says:
“Jus’ wanted to feel that girl’s dress—jus’ wanted to pet it like it was a mouse—Well, how the hell did she know you jus’ wanted to feel her dress? She jerks back and you hold on like it was a mouse. She yells and we got to hide in a irrigation ditch all day with guys lookin’ for us, and we got to sneak out in the dark and get outa the country.
When they arrive, Lennie and George have a conversation about what to do if things do not go well. In other words, George is a realist. He know that things may not go well with Lennie around. So, he plans for the worst.
As the story progresses, it is clear that Curley is going to cause some problems for Lennie. George can read the writing on the wall. And the worst is confirmed when Curley and Lennie fight.
Finally, when Lennie gets into trouble with Curley's wife, he kills her by accident. The men on the farm, especially Curley, plan to get him. George knows that they will kill him. So, George takes Carlon's gun mercifully kills Lennie. Here is the text:
“Yeah,” said George. “I’ll come. But listen, Curley. The poor bastard’s nuts. Don’t shoot ‘im. He di’n’t know what he was doin’.”
“Don’t shoot ‘im?” Curley cried. “He got Carlson’s Luger. ‘Course we’ll shoot ‘im.”
George said weakly, “Maybe Carlson lost his gun.”
“I seen it this morning,” said Carlson. “No, it’s been took.”
Slim stood looking down at Curley’s wife. He said, “Curley—maybe you better stay here with your wife.”
Curley’s face reddened. “I’m goin’,” he said. “I’m gonna shoot the guts outa
that big bastard myself, even if I only got one hand. I’m gonna get ‘im.”
Slim turned to Candy. “You stay here with her then, Candy. The rest of us better get goin’.”