The 31-year-old played against New Zealand last month, but revealed in a social media post he had since had an operation on his shoulder, having already had surgery last year.
Exeter boss Rob Baxter confirmed Slade had initially suffered the problem while representing England during the Six Nations, but that the joint was sufficiently stable for the midfielder to continue playing through last season.
"We were aware that there was a slight issue, as were England, and it has slightly worsened throughout the summer tour," Baxter told the Exeter club website.
"After having the shoulder re-scanned and checking in with how Henry feels about his shoulder, we have made the decision that now is the right time to have the operation.
"Hopefully, he can use this down period to complete a strong rehab stint before we get back into rugby.”
Baxter added Slade would could need up to four months of recovery, so would "miss a good chunk of the start of the season with us".
"Although this is unfortunate, we know he is going to work very hard, and I predict he will still have a very big impact on next season both for us and for England."
Exeter finished seventh in the 10-team Premiership last season, six points adrift of a place in the title-deciding play-offs.
The southwest side begin the 2024/25 campaign at home to Leicester on September 21.
