The Leeds legend was reportedly a target for The Whites when Peter Reid left Elland Road last November.
The Saints announced in a statement that Strachan had decided not to extend his contract beyond the end of this season.
"Gordon and I have fully discussed the situation and are comfortable with his decision," said chairman Rupert Lowe.
Strachan stressed that he made the decision to have the operation after consulting his family when he became Southampton boss.
"It is not a football thing or about money or about the club," he said.
"I made the decision with my family a few weeks after I took the job that I would see out my contract and then walk away.
"I am not going anywhere else and I am not cracking up so people can stop looking too deeply into it.
"This is probably the least stressful time of my seven years in management. The only thing wrong with my health is my hip which needs replacing.
"The chairman asked me to sign a new contract in the summer and he asked me again several times during the last five months.
"I told him in October I would not be signing. He asked me several times to reconsider and he has been absolutely fantastic.
"We hoped to keep it quiet until the end of the season but it was leaking out so we have been forced to say something."