Vaughan and slow left-armer Giles were both forced to return early from the pre-Christmas trip to Pakistan.
Vaughan had his troublesome right knee cleaned out at a Sheffield hospital and Giles underwent surgery in London on a hip complaint.
Both are expected to be included in the England squad to be announced Friday for the Tour which starts next month and will include three Tests and seven One-day internationals. They will need further medical tests before the squad assemble on February 12.
"Two weeks ago I was moving slowly, but I'm progressing nicely now. The real test will be running, turning and twisting with spikes on," Vaughan told the Daily Telegraph.
"If it continues to progress as it has been over the last two weeks, I've no doubt I will be ready for India.
"I will be able to run on a machine in a week or so and then I will go through some cricket skills and exercise at Loughborough on Jan 30 and 31."
Giles is already running and he hopes to start bowling again next week.
"There's still a bit of soreness from the operation but I saw the specialist again last week, he's happy with my rehabilitation and I am confident of making it," Giles added.