The 19-year-old has been an integral part of the Reds' resurgence side over the past year, and has become even more essential following the departure of Luis Suarez to Barcelona in the summer.
The Anfield outfit are poised to offer the England international a five-year deal with £100,000-per-week wages to ward off interest from Real Madrid, who have been monitoring Sterling's progress.
A source told The Mirror: "Talks with Raheem have progressed very well and we don't see any problems arising from this point.
"There is no rush to get the contract signed, but we are now at a stage where we are talking weeks rather than months."
Sterling has made 10 appearances for the Reds in all competitions this season, scoring three goals.