The centre-forward joined Chris Wilder's side on a short-term agreement in January but has now told officials in the Welsh capital he would prefer to stay at Bramall Lane rather than return south as scheduled next month.
Madine, speaking ahead of today's game against Nottingham Forest, said: "This feels like home to me now. I'd be devastated if I had to move on from here at the end of the season. I'd love to stay for as long as the manager wants me and hopefully that's beyond the summer.
"I knew I'd enjoy it here, that's why I jumped at the chance to come, but I wasn't thinking about the future, I just wanted to come snf play games, score goals and win promotion again."
Madine, previously of Sheffield Wednesday, enters the lunchtime fixture searching for his fourth in United's colours as Wilder's team attempts to draw level on points with second-placed Leeds.
"These lads are a special bunch, working daily with this manager and his staff is something I've relished," Madine added. "I don't want to be disrespectful to Cardiff City but it's a long way from home and it's never really happened for me there.
"I'm settled here, I'm content and happy. I've proved I play my best football when I'm happy and I couldn't be happier than I am at Sheffield United."