Michael Palfrey told South Korean media this week he hoped Kim Woo-min won gold instead of Sam Short and Elijah Winnington in the men's 400m freestyle.
The trio - all former world champions - will face off on Saturday in the opening day of action at La Defense Arena.
Head coach Taylor blasted Palfrey as "un-Australian", deeming his actions "extremely disappointing".
But he stopped short of following through on a threat to send him home, saying athlete welfare across the 41-member pool team factored into his decision.
"We took the time to consider the two options - for Mick Palfrey to stay, and for him to be sent home," he said in a statement.
"After extended conversations with athletes, the athlete leadership group, and coaching staff we have made the decision for Michael to stay.
"It is not one we have taken lightly and I stress this is an ongoing discussion for when we return home."
Palfrey works with Australian swimmers Zac Incerti, Abbey Connor and Alex Perkins, but also mentored Kim while he was training in Brisbane ahead of the Games.
According to The Sydney Morning Herald, Palfrey was wearing Australian team colours when he was interviewed by South Korean journalists and declared "Go Korea" while talking about his hopes for Kim.
"I support Rohan Taylor and his team and the decision they have taken in the interests of the swimming team," said Australian chef de mission Anna Meares.
"It is a decision based on high-performance and well-being."
