The 25-year-old defender was signed by Brendan Rodgers in the summer of 2014 from Sevilla after garnering a big reputation in his homeland. However, the full-back has since struggled to establish himself in the Liverpool's line-up.
Moreno had a promising start to the life under Klopp during the 2015-2016 campaign but soon fell out of favour. The Spain international was restricted to only 18 appearances during the whole last season, with his status at Anfield underlined by Klopp's decision to often play midfielder James Milner in Moreno's favoured position.
The Spaniard's situation is unlikely to change in the coming season after last month Klopp decided to strengthen the position further with the arrival of Robertson from Hull City in a deal reportedly worth £8m plus add-ons.
Moreno is happy at Liverpool and enjoyed an impressive performance during the Tuesday's 3-0 win over Bayern Munich after starting ahead of Robertson at the left-back. However, speaking after the game, he admitted that he still may need to leave the club this summer in order to secure more playing time somewhere else.
"Many things have being said [about my future] but right now I am fighting for my position in the left-back. It's true that last season I did not play as much as I wanted but football is like that. One day you are up, another day you are down. This year we start from scratch again. I have to work from the first minute and let's see what happens during the transfer window," Moreno said to Cadena Ser.
"Yes. It is clear [until 31 August everything can happen]. Robertson has arrived, we are now three left-backs. Milner is currently injured. There are many turnarounds in the football. I have to fight and see what happen in the new season.
Questioned whether he would like to return to La Liga at some point, he added: "Well, of course. You always miss your home, your country, the weather, the food... but I am happy here. Let's see what happens this year."
Klopp, however, suggested after the game that Moreno may still have a chance to resurrect his Liverpool career after praising his impressive display against Bayern.
"If he plays like he did today then yeah (he has a chance of playing)," Klopp said in the press conference. "It was a really good game from Alberto, that's why we kept him on, because every second he was playing better, he was more present."
"He's really fit but he's had a little problem with his thigh since the day before yesterday. But we had a test with him this morning and he really played well. The players decide themselves how many games they play."