The 26-year-old sustained the injury during the Heineken Cup quarter-final defeat by Saracens earlier this month and after seeing an ankle specialist, will have surgery on Wednesday.
A six-week lay-off would mean Henson missing the June 7 clash in Bloemfontein and June 14 in Pretoria, while he is certainly out of the Ospreys' three remaining Magners League fixtures.
Ospreys physio Chris Towers said: "Gavin's injury hasn't responded to treatment as anticipated and so he was routinely reviewed by an ankle specialist.
"In light of the ligament damage involved the specialist felt that it was advisable at this stage to examine the potential for any instability of the joint and in this instance, the only way to achieve this is to perform an arthroscopy and screen the ankle in theatre.
"The best case scenario will see Gavin sidelined for a further four to six weeks, with no further action needing to be taken.
"However, if the examination reveals clinical instability then the plan will be to repair the ligament at that time, which would be likely to sideline Gavin for approximately 12 weeks.
"It is impossible to say what the outcome will be until the specialist has had a chance to assess the ankle fully, so it would be inappropriate for us to comment further at this moment."
The news will come as a huge blow to both Henson and Wales head coach Warren Gatland, particularly with doubts remaining over Mark Jones' availability for the double-header.
Henson missed the World Cup last year after struggling with an Achilles problem which wrecked his preparation for the tournament in France.
However, the centre returned for Wales' triumphant RBS 6 Nations tournament earlier this year and started every game of the championships.
Ospreys head coach Lyn Jones said: "The news comes as a blow for the Ospreys as we look to finish the season in winning style, and personally for Gavin, who has been in such good form this year.
"Hopefully, Wednesday's investigation will bring him good news and he can be back up and about within just a few weeks."