The 32-year-old has had to take a watching brief this season after severe bruising on his knee led to fears the injury could develop into a stress fracture.
Britton has now undergone further scans that have shown signs of improvement and is set to start running again within the next fortnight.
The midfielder is now focused on returning to action next month.
"It is getting there slowly. The scan results were positive - the bruising on the bone is going but it's not completely gone yet," Britton told Swansea's official website.
"I have to do some treadmill work and hopefully I will be out on the pitch running in a week or two.
"It's the longest period I have been out since I left school. It sounds crazy but I can't remember anything since I had a stress fracture in my back when I was 14.
"That kept me out for six months, but since then I can't remember anything that was more than a month maybe.
"You can't rush things like this. I have been doing the oxygen chamber, seeing if there are any injections or if there is anything weird and whacky we can do. I'm trying everything to speed up the recovery, but unfortunately you just have to wait.
"We are hoping maybe mid-November. I have another scan coming up in a couple of weeks when I hope to get the green light to train properly."
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