Nobody gave the paw to Mr. White. He grabbed it off the fire himself when Morris threw it there.
Morris threw it there because he did not want anyone else to have it. He knew that any wishes that someone made would be granted, but they would probably be granted in a very bad way.
Because he knew that, he wanted to just burn the paw and be done with it (I don't know why he hadn't just thrown it away before). But instead, Mr. White grabbed it.
Just as Morris predicted, the White family soon wished that Mr. White had not gotten the paw and had not made any wishes.
In the short story "The Monkey's Paw" the Whites and Sergeant-Major Morris are sitting by the fire talking. Mr. Morris is sharing a tale with the White's about the monkey paw. He is telling them how it has the power to give three wishes. Mr. White is intrigued. Morris is concerned because he knows that the monkey's paw has brought nothing but grief and tragedy to those who have made wishes using it. He tosses it into the fire so he can end its ability to harm anyone else. However, Mr. White is excited and curious if the paw actually works. He grabs it off of the fire so that it will not be destroyed so that later on he can test its abilities.