England play their tournament opener against Scotland under new head coach Steve Borthwick at Twickenham on February 4th, before hosting Italy eight days later.
Chiefs rugby director Rob Baxter described the injury as "relatively bad".
Cowan-Dickie, who was hurt during the second half of Exeter's Premiership victory over Northampton on Saturday, has not travelled to South Africa with the Exeter squad for Saturday's Champions Cup appointment with the Bulls.
"I believe he is seeing a surgeon today and the surgeon will then give us an indication," said Baxter. "I think at this stage, we have got to say it is relatively bad.
"I don't want to be flippant and say it will be eight weeks, 10 weeks, 12 weeks (out).
"I think we are assessing that, but you are not going to be seeing him running around in the next three or four weeks."
Cowan-Dickie, who has won 41 England caps, is in his final season at Exeter before joining French champions Montpellier.
Chiefs' Scotland full-back Stuart Hogg has also not made the journey to Pretoria as he battles a heel problem.
"Stuart did a fitness test on Saturday pre-game and still couldn't run at speed or couldn't put any distance in his heel. It was still too sore," Baxter said.
"He will be doing a kind of rehab programme this week while we are away. As soon as his heel is comfortable enough, he will be declared fit to play again. It's not like a ligament thing or something that will require an operation."
