In the comprehensive defeat Liverpool showed familiar defensive frailties after failing to bolster that area of their back line during the summer and instead bringing in attacking players Mohamed Salah and Alex Oxlade-Chamberlain, both for significant fees.
Klopp targeted Virgil van Dijk earlier in the summer, but the Reds withdrew their interest and issued a public apology to Southampton after the Saints complained to the Premier League over an alleged illegal approach.
The Daily Telegraph correspondent Law said: "Didn't the result tell us more about Liverpool's defending rather than Manchester City's attacking?
"It showed they don't have a squad capable of shutting up shop - they haven't addressed problems already there.
"I wasn't impressed with them at Watford - they've conceded eight goals in two away games. I find it weird that Jurgen Klopp is going for players like Alex Oxlade-Chamberlain and Thomas Lemar, yet it's absolutely clear they need a central defender and possibly a goalkeeper.
"I don't get why they made Virgil van Dijk their number one target and then when they couldn't get him, say that they're not interested in a central defender. There are other central defenders out there that would improve Liverpool's defence."