The 21-year-old Saints left-back has been limited to only eight appearances for Rovers since joining on an emergency loan in October, with an ankle injury keeping him sidelined for six weeks.
However Mills, who represented England at Under-17 and Under-18 level, made his return in the 3-0 win over Scunthorpe on Saturday in which he netted his first goal for the club.
And he told the Sheffield Star: "I've been playing games here and I've been enjoying my football so it would be great to stay on. But obviously it is not up to me. It is up to the manager here and the manager at Southampton.
"I'm just looking to keep playing well and keep working hard and see what comes about."
Source: PA
Source: PA