The Brazilian sustained the injury after landing heavily following a collision during the Reds' Premier League clash with Swansea, and is expected to be out of action until the end of October.
But the player himself has now revealed his treatment is going well, and insists he will do his best to return to action earlier than planned.
"It's annoying to get injured, especially when you are playing well, but that happens," Coutinho told Globo Esporte.
"I now want to focus on my recovery and do everything right to be back as soon as possible to help Liverpool in the league.
"The surgery was a success and this is what matters the most. I'll be back soon."
Coutinho will be in contention for Luiz Felipe Scolari's Brazil squad for this summer's World Cup if he can maintain his promising start to life in Liverpool, and admits a place at the tournament is his dream.
"I dream of getting called up again for the national team, but it's only up to me," he added.
"Of course it's bad when you get injured in a moment like this, but this is football.
"I have had a set-back and my concern is being fully ready as soon as possible. If I play well, things will happen naturally."