However, with his current deal tying him to Liverpool until 2022, and the Reds boss receiving significant financial backing during the recently closed transfer window, it seems Klopp plans to remain with the north west outfit for the long haul.
There are more hard things to accept in life [but] it's August and September and then we start playing three games in a week and then we don't have that problem any more pretty quickly. Let me say it like this, previous year he already had a few problems with that but he could obviously play - and played really good.
"We have to invest more, we have to fight more, we have to create more".
Jurgen Klopp will be extremely happy with the result and the team's performance, who were by no means exceptional but never looked like they weren't in control of the match.
Pellegrini conceded that West Ham need to do better.
"There would always be a point where we spent a bit more money". So to sign someone of Alisson's calibre will definitely turn out to be a major coup in my opinion, regardless of the price tag, and it could be the difference between you winning a trophy and not winning one this season.
Liverpool's midfield provided the foundation to both scorch and stifle with Gini Wijnaldum controlling well at the base, while Milner was core to every facet of play and Keita exposed spaces the visitors knew nothing about until he weaved through them.
The Egypt global scored 44 times in his maiden campaign, which ended in tears with his premature exit from the Champions League final because of a shoulder injury which contributed to the defeat to Real Madrid, but normal service was resumed within 19 minutes. The Englishman's work ethic and overall play was hardly curtailed by the thick bandage around his head, concealing 15 stitches.
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West Ham will continue their search for their first win of the season at the London Stadium against Bournemouth on Saturday.
"It's a similar style of play and a similar position that he played at Leipzig-he can play different positions, but I think the No. 8 is his best in the moment". The challenge now is whether they can take the extra step and secure silverware.
What's more, he's scored on the opening day in each of the last two seasons so loves to hit the ground running.
One Twitter user, AnfieldRd96, didn't hold back, saying "I'm in love with Naby Keïta".
Therefore, steering clear of that this year, playing some attractive football and ending with a comfortable top 10 finish along with a good cup run would probably be enough to appease many of our fans, with the ambition to progress season upon season from there.
During the previous two summers, first in California and then Hong Kong, the manager addressed the psychology of his squad and the wider Liverpool world.
"I feel as though we can form a really tight group, working together to support each other".
Of course, it is easier said than done though.