The 27-year-old Samoan, in his second season with Worcester after arriving in 2019, must now undergo surgery after suffering the injury in training.
"The early prognosis is that he is likely to miss the rest of the season," said Warriors boss Alan Solomons.
"Melani suffered an unfortunate hamstring injury in training which will require surgical repair."
It is the second major injury suffered by Nanai, who has been limited to just 17 appearances - scoring three tries - since his arrival from New Zealand Super Rugby side Blues.
He suffered a shoulder injury in February 2020 which also needed surgery.
Nanai was ruled out for the rest of the original season then too, but the delay caused by the Covid pandemic allowed him to return when the Premiership campaign resumed in August.
In his first game back against Gloucester, he was sent off and subsequently banned for three games for a no-arms tackle on England winger Jonny May.