The Reds have never lifted the Premier League trophy with their last title success coming in the 1989-90 season when winning the old Division One crown.
Jurgen Klopp's side went very close last term when they finished second, edged out by just a point by Manchester City in a thrilling title race, and they are expected to battle for the top domestic prize again this season.
They have made a fantastic start and sit top of the table with four wins from four so far - the only team in the top flight with a perfect record - and look in good shape as they aim to become champions for the first time in 30 years.
Last season's Champions League success has only served to whet the appetite for more trophies at Anfield and Wijnaldum admits they want to collect more silverware but Klopp is not putting the squad under pressure to deliver.
He told the Liverpool website: "Now we know what it's like to win the biggest trophy of them all - and we just want to keep going from here and add even more titles.
"But Jurgen Klopp is not putting extreme pressure on us to win the Premier League. He does not tell us, 'Now you must win this or that trophy'. He is very relaxed. He did not do that during the past season, either.
"His idea is that we get the maximum out of every individual every single day that we are together. And then we will see where that brings us at the end of the season."
Wijnaldum has enjoyed a good international break as he scored and got an assist for the Netherlands in their 4-2 win Euro 2020 qualifying win in Germany on Friday and will be expected to feature tonight against Estonia before returning to Liverpool.
The midfielder will then switch focus back to club duty with the Reds taking on his former employers, Newcastle, in the Premier League on Saturday.