Swansea manager Michael Laudrup signed the England youth international from Arsenal towards the close of the transfer window for just under £1m.
Bartley only managed one appearance for the Swans in a Capital One Cup tie before suffering the thigh problem.
"I've reached the five-week stage today, I've been for a scan recently and I was told it was coming along well," he told the club's official website.
"I've got another eight weeks of recovery ahead of me, so I am getting there. It wasn't too bad in the first couple of weeks, but then you're out of the group and in the gym on your own a lot.
"But I'll continue working away in the gym, along with receiving massages and physio work to help my recovery. I'll have another scan in a couple of weeks and hopefully that will be another good round of news."
The 21-year-old is fondly remembered by Rangers fans after impressing in two loan spells at Ibrox, playing regularly in central defence and an unfamiliar defensive midfield role.