The 37-cap forward suffered the setback just five minutes into his country's 44-15 win over Argentina and was immediately replaced.
Although the exact nature of the problem has not been confirmed, the Warriors says he has had surgery on his leg and will now miss the next six months.
"Tim Swinson has undergone surgery on a leg injury sustained in Scotland's 44-15 win over Argentina in Resistencia on Saturday,' a short statement on the club's website said.
"The 31-year-old returned home to undergo the operation and is expected to be out of action for around six months."
It means Swinson will miss at least the first half of the Pro14 season, while he is also a major doubt for next year's Six Nations, a problem that could also lead to him falling down the pecking order ahead of the 2019 World Cup.