O’Driscoll, 33, was forced off midway through the reigning Heineken Cup champions’ 59-22 victory at the RDS in Dublin earlier this month.
Following scans and a specialist review by the Ireland medical team, it was determined the Ireland captain will be sidelined for eight to 12 weeks.
With Rory Best, Stephen Ferris, Rob Kearney and Sean O’Brien also missing through injury, Sexton is being touted as a possible skipper during the Guinness Series.
“A fly-half’s role is to lead the team anyway. Goalkicking and punting are just a part of his job,” Tainton said.
“I wouldn’t see it being a burden to take on the captaincy as well.”
Munster veteran Paul O’Connell is the favourite to take over the armband, and Tainton revealed the 33-year-old and Cian Healy are expected to be fit for the opener against South Africa.
“Paul trained fully today and did everything in training – scrum, line-out and contact,” he added.
“We’ll see what happens over the next 24 hours but we don’t envisage any reaction to what he did today. It’s looking good from his side.
“Cian did a full session today and there were no problems with his shoulder during a live scrummaging session. He showed full power.”
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