The 23-year-old arrived from Southampton on a two-month emergency loan deal last week as Rovers manager Sean O'Driscoll responded to the latest spate of injuries to hit his squad and Gillett wants to make the move long-term.
He told the club's official website: "Yes I think so, it's something I'd like to do if things go well and the club want me to stay I'd be very happy to do that.
"I didn't start the season looking to come out on loan, I thought I'd have a chance to play at Southampton after playing a few times last season.
"But things change in football. There were a few clubs interested but Doncaster were by far the best choice.
"I'm a central midfielder so I like to get stuck in, break up play and get on the ball. I'm told Doncaster are a really good footballing side and I hope to fit in quite nicely."