The 20-year-old, who is currently on loan at the club from Liverpool, is due to return to his parent club at the end of December.
However, the talented midfielder is open to the prospect of spending the rest of the campaign with the League One club.
"I'm very happy here. It's a fantastic club. All the lads and the staff have made me feel very welcome and the fans have been great too," he told reporters at a press conference.
"I'm enjoying every moment at the club. The move has been really beneficial for me. I feel I've developed by playing regular first-team football.
"If staying all season is an option, it would definitely be something I would look at, but that's down to both clubs really. Right now, I'm just concentrating on getting my head down, playing well and we'll see what happens."
Coady has appeared 15 times for Sheffield United this season.