Coutinho joined the Reds back in 2013 from Serie A outfit Inter Milan and made more than 200 appearances for the club before leaving in a big-money move.
Liverpool were linked with a move for the playmaker, who failed to impress on his dream switch from Anfield to Barcelona.
Klopp admitted it was a possibility that Coutinho could have returned to Merseyside but the Reds did not have the budget for the player, while they are already blessed with the forward trio of Sadio Mane, Mohamed Salah and Roberto Firmino.
The former Borussia Dortmund manager actually went on to aid his former arch rival club Bayern by suggesting to Coutinho that a move to the Bavarian giants would be a good one over the summer.
Klopp told German newspaper Bild: "I personally told him that Bayern and him would be a good fit.
"He's a world class player and a really good signing for Bayern - and the Bundesliga."
Klopp has been linked himself with a move to Bayern Munich in the future and his recommendation of the club to Coutinho will have certainly done him no harm with the Bundesliga champions.
The Reds manager has previously admitted he could be tempted to take a break after his contract at Anfield comes to an end in 2022 and the Bayern job remains a possibility if and when decides to return to management after his time with Liverpool.
Meanwhile, Liverpool will be looking to maintain their perfect start to the Premier League season when they welcome Newcastle United to Anfield on Saturday.
The Reds are sitting pretty at the top of the table after winning all of their first four games.