Hinshelwood, who is enjoying an excellent breakthrough season, will require surgery.
The 18-year-old made his senior debut last season, but this term has become heavily relied upon due to the Seagull's injury crisis.
A midfielder by trade, Hinshelwood already has three goals in just 12 Premier League appearances, playing mainly in defence.
Manager Roberto De Zerbi has had to contend with a number of big injuries this season, namely long-term issues for starting XI regulars Julio Enciso, Solly March, Ansu Fati and Pervis Estupinan.
Problems have eased in recent weeks, but the Italian coach will now be without Worthing-born Hinshelwood for the away trip to Tottenham.
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De Zerbi, meanwhile, will serve a touchline ban at Spurs on Saturday.
The combustible Italian was shown a third yellow card of the season by referee Simon Hooper during Sunday's 4-1 Premier League win over Crystal Palace.
De Zerbi felt Palace debutant Daniel Munoz handled as he attempted to prevent a Brighton attack after just seven minutes and was booked for his reaction.
But the 44-year-old believes his squad will not be affected by his absence from the dugout.
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"The yellow card was right," said De Zerbi. "I forgot that with another yellow card I can't be on the bench at Tottenham - but my players can play without their coach there."