Klopp suffered his heaviest defeat as Liverpool manager as his team was taken apart in ruthless fashion by their Premier League title rivals at the Etihad Stadium.
The German coach had opted to make changes to the team that swept aside Arsenal 4-0 at Anfield before the international break, with Trent-Alexander Arnold and Ragnar Klavan coming in for Joe Gomez and Dejan Lovren, respectively.
Thompson raised questions over the need for changes to a winning starting XI and also took aim at Klopp's failure to sufficiently address his side's defensive issues in the transfer market.
"I still think we could have strengthened, without a doubt," he told Sky Sports. "We did sign Andrew Robertson, but he didn't play against City or Arsenal, so yes we should have signed somebody.
"But for me, the selection was the big issue. After such an impressive result there is no reason to make changes because it's such an important thing for a defence being settled, especially against such an offensive team when you've got to trust your teammates.
"Lovren was on the bench and I always think if you are on the bench then you should be fit enough to play, so I can't understand why Klavan played.
"Alexander-Arnold is a fantastic offensive player but even Klopp has questioned his defensive work, so I was expecting the same back four with Gomez in there."
The defeat against City was heavily influenced by Sadio Mane's sending off late in the first half for a dangerous challenge on City goalkeeper Ederson Moraes, but Thompson insisted that Liverpool should have performed better even with 10 men.
"We should not get beat 5-0, 10 men or not. I thought we gave in and gave up and no Liverpool team should be like that," he said.
"If you are organised and disciplined then you can still make it extremely hard for the other team and we didn't do that."