It is getting more interesting as time goes on that Liverpool doesn't make any transfer at all... I mean we are only 10 days behind the 1st week of the season in the PL. So when do they plan to sign someone?

Yes, none of their signings have been made, apart from the Liverpool U21 players themselves, and they are still just ordinary players. There is only one rumor that is always associated with Liverpool, namely Barış Alper Yılmaz, Right Winger from the Galatasaray club. That's all until now there has been no progress.
Source: Liverpool FC Rumours about signingsI will love to see what decision Julian Alvarez finally takes after returning from the Olympics. It's either he moves of becomes a second striker yet again behind Erling Haaland, nothing will change.
Atlético Madrid -> Julián Álvarez deal!Finally, Atletico Madrid dares to spend €75m to get the Man City Center-Forward. It's just a matter of time until the signing is done. Great job Atletico Madrid.
Source: Atlético Madrid are closing in on Julián Álvarez deal!