James Milner believes Liverpool have earned the respect of Europe with their Champions League rise under Jürgen Klopp but admits he will not be satisfied until a trophy finally returns to Anfield.

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Liverpool’s comfortable defeat of Bayern Munich at the Allianz Arena, where the serial German champions had lost only one of their previous 26 Champions League games, was arguably their finest European away display under Klopp and demonstrated the quality that delivered a place in last season’s final. England will have four representatives in Friday’s quarter-final draw and, though keen to avoid an all-Premier League affair, Milner believes all of Europe view Liverpool as a team to avoid.

“I’d like to think teams have that respect for us,” the midfielder said. “We got to the Champions League final last year. We got to the Europa League final in 2016. There are two European finals since I have been at the club. We are challenging for the Premier League too. Whether teams take notice is not really for us to worry about but they take notice after they have played us and we have put them out.”

Liverpool are back on the international map, according to their manager, but Milner admits the failure to win a trophy under Klopp is hard to stomach. It is also a motivation to go one better in this season’s Champions League and for the Premier League title race, the 33-year-old says.

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“I have been to three finals since I have been here, lost them all and I am desperate to put that right. It’s a memory you never forget. You are proud to have played in the European Cup final but you don’t go there to lose. I have loved every minute of my time at Liverpool but to have not won something when we have given ourselves opportunities is difficult to take.

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“It is something I am desperate to put right. It’s been so long since we won a league title. Even in recent years Liverpool haven’t won as many trophies as they should do for such a big club. We have to put that right. We have to get over the line in one of them.”

Georginio Wijnaldum, a fellow midfielder, shares Milner’s assessment that defeat by Real Madrid in last year’s Champions League final has increased Liverpool’s desire for a sixth European Cup. “We felt what it is to play in a final and the journey was great. We want it again,” he said. “At a club like Liverpool you have to win everything possible and you have to play for both: the Premier League and the Champions League. It suits Liverpool to play for both. Look at Manchester City, they are not [just] going for the title or the Champions League. It is a big club – so is Liverpool – so you go for both.”

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Milner insists Liverpool can improve on the performance that ensured there would be no German club in the last eight of the Champions League for the first time in 13 years. “We have played better games but you could see some of the experience we have gained over the last few years come through,” he said. “Sadio [Mané]’s first goal was very naughty. The way he took it was ridiculous but that’s the quality he’s got.”

Milner believes the strength of the Premier League is finally beginning to translate on the European stage, with Manchester City, Manchester United and Tottenham joining Liverpool in the quarter-finals. But the prospect of another all-English tie does not excite him.

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“We have always had the belief that the Premier League is the best in the world but it has been a top league without that many teams [in the latter stages] of this one,” he said. “Every team left are a good side. People will look at Porto and Ajax but they are young teams and they are flying. They deserve to be there and we all know they will be tough teams. To have four teams in the last eight is great for England but I don’t particularly want to play any of the others.

“It is always a weird feeling playing an English team in Europe. You get used to getting on a plane and instead you end up getting on a bus and going across the M62. Whoever we get will be tough. But we have shown over the last few years what we can do and hopefully we can keep putting in performances and kick on.”

Liverpool’s win also dashed Pep Guardiola’s hope of seeing Bayern, his former club, in the last eight. “Ah, yes,” said Milner, well aware of the City manager’s wish. “That’s unlucky for him, isn’t it?”