Sturridge has struggled for game time under Jurgen Klopp due to a combination of form and injuries, and spent the latter half of last season out on loan at West Brom.
However, he has arrived back fit and fighting this season and has been impressive off the bench, including scoring a late equaliser against Chelsea.
"I think Sturridge is a fantastic player. When he went to West Brom he may have realised that 'I want to be back at Liverpool'," Rush told the 'Mirror'.
"He's got a second chance now and he's done really well so far. Hopefully that will continue. As for a new contract? Yeah, I think so.
"I think it's one of those where he's playing for a contract now. So it's even more important to him to show how good he can be."