The former Manchester City youngster has entered the final six months of his United deal, just as he has established himself as part of United's squad after joining from City in 2018.
Bryan made his latest Premier League appearance against Chelsea at the weekend, after scoring his first goal for the Blades in the victory at Manchester United last month.
And when asked if talks had begun over extending his contract, Bryan said: "No, I'm just enjoying my football at the moment and trying to give my best to the team. Of course I'd like to stay.
"It's taken longer than I would have liked [to get a chance], but I was behind Jack O'Connell, who was someone I looked up to when I first came in - a big character and a fantastic player.
"I understand football and the rhythm of the back line in the promotion season was settled.
"I went out on loan for some game time to get the happiness of football back, and I came back in pre-season raring to go. I got my chance and it's up to me to take it.
"Every player will say the same you're not happy if you're not playing. But I'm not focused on my situation, I'm just here to focus on my football and give everything for Sheffield United.
"There's a lot of football left to play so we just need to keep fighting."