Gallagher, 28, has started in all six of the Blades' games since arriving for an initial month in the middle of September and Wilson is keen to keep hold of him after Leicester boss Nigel Pearson confirmed he is happy for the Scotland international to stay.
"I'm hoping it will be done before the weekend," Wilson said. "I think Paul's starting to show exactly why we brought him here in the first place.
"He was a little bit rusty early on in the games with us, but now he's settling into the role that we'd like him to play and get an understanding of the players around him, so the more games he plays, it's quite obvious now, he'll get better.
"We'll see now why he's highly thought of at Leicester and certainly with us. He's a goalscorer, a striker, so he's always got that threat of a goal.
"You know he's always capable of popping up at the back post or coming inside and creating or scoring himself, so it gives us another option as to where the goals are going to come from rather than putting the pressure on the strikers all the time."