French media reported that the former Paris St Germain and Liverpool centre-back allegedly had an altercation with manager Michel Der Zakarian.
Sakho, capped 29 times, moved to Montpellier from Premier League side Crystal Palace in 2021.
"You have to know how to leave the table when respect is no longer served," Sakho said on Instagram.
"Following the incident which took place last week at the training centre and for which I decline all responsibility, I decided... to end my collaboration with Montpellier."
Montpellier said the two parties have decided to end the contract after 10 days of reflection.
"For ten days, a media frenzy has been created around the club around a 'clash' between Michel Der Zakarian and Mamadou Sakho," Montpellier said in a statement.
"Contrary to what has been said and relayed, no precautionary dismissal has been taken against Mamadou Sakho."
Sakho also fell out of favour at Liverpool in 2016 when he was dropped from the first team and was sent home from a pre-season tour of the United States for disciplinary reasons.
Montpellier, sitting 11th in the French top-flight, visit second-placed PSG later on Friday.
