Former Leeds forward Beckford, 28, has scored three goals in six appearances for Huddersfield since joining on loan from Leicester in September, initially until January 1, including the winner at Barnsley on Saturday.
"I think Jermaine's view is he'd probably like to stay with us for the season and hopefully that can be arranged," Hoyle told BBC Radio Leeds. "He loves it here and gets on very well with Simon Grayson."
Huddersfield sold Jordan Rhodes to Blackburn for £8million on transfer deadline day and manager Simon Grayson targeted Beckford as his replacement having worked with the forward during his time as manager at Leeds.
Hoyle added: "Jermaine is a fantastic character. Before I met him I thought he had a bit of an attitude, but since I met him he's been a cracking guy. We'd like him to be around longer term, but ultimately it's Leicester's call.
"I never thought that anyone would come in and pay £8million for (Rhodes) and the timing of the deal was an absolute nightmare in terms of getting a replacement."