They were linked with a move for new Chelsea loanee midfielder Denis Zakaria and eventually ended up signing the Old Lady's Arthur.
The Brazilian has joined on a season-long loan with the option to buy for £32 million.
However, Liverpool were also reportedly in for Laimer, who was coming off the back of an impressive 2021-22 campaign with Leipzig.
The Austrian made 43 appearances, finding the net on three occasions whilst contributing five assists.
The midfielder has intriguingly revealed that he is a fan of the Merseysiders as he touched on speculation of a move to Anfield (via Bild):
“I’m generally a football fan. I generally like watching the Premier League. I’ve always been a Liverpool fan."
“Surely there are such thoughts. I don’t think a club like that comes knocking every summer, it’s one of the best in the world. I’m a here and now focused. If you worry too much, you lose sight of the most important thing – and that is playing football and getting better."
Bundesliga giants Bayern Munich made a move for Laimer in the summer but were unsuccessful in their attempts to prise him away from the Red Bull Arena.
“I’ve been at Leipzig for five years and I feel good. Now it will be a year longer and then we will see what happens.”
Laimer has a year left on his contract with Leipzig and is currently nursing a ligament injury.
He is expected to return in January, at a time in which he will be free to speak to clubs from abroad as his contract expiry date looms.
Liverpool's potential pursuit of Jude Bellingham may scupper plans to sign Laimer
Liverpool are being heavily linked with a move for Borussia Dortmund midfielder Bellingham.
However, reports claim that Real Madrid are the current frontrunners for the English teenager as the Spanish side are the club he wants to join.
Bellingham has impressed for Dortmund, making 100 appearances, scoring 13 goals and providing 18 assists since arriving from Birmingham City back in 2020.
He has three years left on his contract and Liverpool continue to be an interested suitor.
That may jeopardize any plans Laimer has in joining Klopp's side unless the German coach decides to oversee a major overhaul of his midfield.