The Reds are currently second in the Premier League, four points behind league leaders Chelsea, and they have a six-point buffer to Tottenham in fifth place.
Liverpool have not featured in Europe's premier competition since the 2009/10 season when they were knocked out in the group stages.
Suarez, who is contracted with the Merseyside club until 2018, admits he is settled at Liverpool but would not rule out playing in Italy one day.
"A future in Italy? Never say never," Suarez told the Gazzetta dello Sport.
"For a South American player who wants to get into history, it is advisable to have an experience in Serie A.
"Italian football is a top level and I follow it assiduously. Two years ago I received an offer from Serie A, I don't want to name the team, it was a difficult time for me, but after thinking about it I decided to stay in Liverpool."
The Uruguayan striker has contributed 24 goals for Liverpool in 27 appearances this term but revealed he is eager to sample an Anfield Champions League experience.
"And now things are going perfectly for me," he claimed. "We are fighting for the title but above all we are very close to qualifying for the Champions League.
"I cannot wait to finally play the big European matches at Anfield. I could not do it in these three seasons and when my team-mates tell me about the atmosphere of those nights it gives me goose bumps.
"This year we can achieve it, I feel we'll do it. And I don't lose the club since I have recently extended my contract."