The Reds legend believes his former side can go all the way after proving that they thrive in continental competitions.
Jurgen Klopp's men ran deserving and relatively unchallenged 3-0 winners against Villarreal at Anfield on Thursday night to seal a 3-1 aggregate victory in the Europa League semi-finals and progress to a first European final in nine years.
After he took over from Brendan Rodgers in October, Klopp's first season at Anfield will be defined by the European campaign regardless of whether the Reds can beat defending champions Sevilla in Basel later this month.
The semi-final first-leg defeat to Villarreal was Liverpool's only defeat in the 14 fixtures they have played in the competition so far this season, and Rush believes the club have proven that they can become a true European force again.
The club's record goalscorer said, as quoted by BT Sport: "Nothing ceases to amaze me at Anfield. It's not a one-off, and now they're in the final there's no reason they can't go on and win it.
"It will be an amazing feat for Sevilla to win it for a third time in a row, but Dortmund were the favourites to win the competition when Liverpool beat them. Liverpool have shown what they can do and Europe is made for them.
"The Europa League has come as a bonus and getting a place in the Champions League would help them attract players next season."
Liverpool will face Sevilla in the final in Switzerland on Wednesday, 18 May, but before then they face Watford, Chelsea, and West Bromwich Albion as they look to continue their push to finish in the Premier League's top six.