The 29-year-old returned to Rugby Park earlier this year on a short-term contract and scored three goals for Killie.
Johnston is busy preparing his squad for Saturday's opening Scottish Premiership game against Aberdeen, and he would like to have the Boyd deal concluded ahead of the trip to Pittodrie.
"Hopefully," said Johnston. "We've been trying to do that for a wee while. I think Boydy's keen to stay as well. And somebody like that can only improve us.
"He's a top quality player. He's played at the highest level and scored goals at the highest level. There are not many players who can do that."
Johnston has some other targets in mind as well, but expects to add those later in the transfer window.
He added: "I don't know if we'll get them in before Saturday. But before the end of the transfer window we're hoping to bring in a few more players."