The 20-year-old midfielder, who joined the Blades for an initial six-month period from Liverpool in July, made his first start under manager Nigel Clough in the weekend FA Cup win at Colchester.
"I'm very happy here," the former England Under-20s skipper said.
"It's a fantastic club. All the lads and the staff have made me feel very welcome and the fans have been great too.
"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 started in 11 of his 15 appearances for the Blades and showed his versatility at Colchester on Saturday by playing in an unfamilar role on the right side of midfield.