The Merseyside outfit showed good form towards the end of last season, losing just one of their final 12 games to eventually secure a seventh-place finish in the top flight.
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"It's got to be the aim [to finish in the top four]," Gerrard told reporters after his testimonial match against Olympiacos. "I've experienced it many times here at at Anfield (European football) that's where we believe we belong.
"[Aspas] got little bits of magic and he'll certainly strengthen the squad, he's still very young but you can see the talent there.
"I thought [Kolo] Toure looked very assured at the back, [Simon] Mignolet kept a clean sheet. And Luis Alberto is one for the future. I'm sure every minute he gets here at Anfield it'll be a great experience for him.
"All our new signings are settling well and hopefully there's one or two more in these coming weeks."