The 28-year-old missed most of last season and the start of the current campaign because of the problem but returned to make 13 appearances for club and country since September.
His last match was an 89th-minute substitute appearance in United's 4-3 win over Newcastle on December 26.
Strachan – who has appointed Mark McGhee as his assistant – said: "On behalf of the Scotland supporters, the national squad and backroom team, I wish Darren the best of luck in his recovery from the surgery that has been confirmed by his club.
"I spoke with Darren on Tuesday and know that the decision to have the operation has been taken in the interests of his long-term health and with the full backing of the medical staff at Manchester United.
"In the meantime, I have stressed to him that, as captain, he remains an integral part of the international set-up and that he is welcome to join up with the squad at any time during his recovery."
A Manchester United statement said: "We wish to stress this was a planned procedure undertaken at the optimal time having achieved a period of sustained good general health, as illustrated by the fact that Darren has trained with the team regularly this season and been available for selection right up to the operation.
"We look forward to him returning next season with this problem firmly behind him."