Jones was pictured on Instagram earlier this week with his arm in a sling, and Glasgow head coach Franco Smith confirmed the 30-year-old had undergone surgery.
"He's got a finger that's given him problems," Smith said ahead of Glasgow's URC meeting with Cardiff. "He got hurt during the Test match at the weekend. The best way forward currently was to have an operation on it.
"He'll be out for the foreseeable future, but we'll take it week by week."
There was more positive news on Jones' centre partner Sione Tuipluotu, who missed the last two rounds of the Six Nations having picked up a knee injury against England.
Tuipulotu was expected to miss up to three months, but Smith hinted he is ahead of schedule.
"Sione's training well, so we'll just take it one week at a time and, as the other guys are playing well at the moment, it's a good opportunity also for them to mentally reset.
"They'll be back when we need them most."