Keane has made four appearances for Derby since joining last month and his deal expires after the New Year's Day game against Wigan Athletic, where his brother Will is currently on loan.
The 20-year-old is enjoying his spell with the Championship club and would be happy to stay, if he will be playing regularly for Steve McClaren's side.
"I would definitely be interested in staying," Keane is quoted in the Sunday Mirror. "Obviously, we will have to wait and see what United think.
"It will also be important to see if I am playing games because that is the most important thing for me.
"But I have enjoyed myself here and if that is still the case and I keep playing football then I would love to stay.
"There is a really positive atmosphere here with all the good results and so much talent in the squad that it is a pleasure to be part of."
Derby boss McClaren has acknowledged that he is interested in keeping hold of the centre-half until the end of the season.
McClaren said: "We have said that from day one and, hopefully, we can. He has proved his pedigree."