Karius, a £4.7million signing from Mainz over the summer, sustained the injury in a friendly in late July and was at that point ruled out for eight to 10 weeks.
But the 23-year-old German could be in competitive action for his new club sooner than had been expected.
Karius said on Liverpool's official site: "I had surgery on Wednesday, so now the wires (the support that had been placed in his broken hand) have been removed that they put in four weeks ago.
"Now the fracture has healed, hopefully I can start training and build it up.
"I just need to see how it feels. But I can now start using my hands and catch balls. It's up to me now, basically.
"We're ahead of schedule because in total it's been about five weeks I think, but I don't want to rush it too much. Now I am in the last stages, I just want to feel good and confident about it, that it's all done.
"I am not going to rush it because of two or three days, but if I feel good then maybe I will be back way sooner than expected."
Despite his injury, Karius has been working hard at Liverpool's Melwood training ground with goalkeeping coach John Achterberg.
And he added: "I don't really think I am going to need a long time to be back on my best level. My body is there, it's ready, it's just the one hand I need to be sure of now and that it's not giving me any problems.
"I am pretty sure next week I will feel fine. Obviously I can't say 100 per cent because I'll need to see how it is when I am in full training next week, so we'll see, but I don't think it is going to be long now."
Source: PA