Saints were understood to have agreed a fee of around £800,000 with City for the centre-half, but the deal is now off.
However, Fontaine's agent Tam Byrne has denied reports claiming the plug was pulled due to a problem with the player's fitness.
"The decision to not proceed at this time was reached amicably by both parties after extensive talks with Southampton FC," he told The Echo.
"I am not at liberty to disclose the nature or content of those discussions but I can confirm that the decision was in no way related to any long-term concerns over his fitness.
"He has appreciated this opportunity to discuss his career with Southampton, however now this decision is reached he is fully committed to returning to Bristol City."