The family had more than 1,000 slaves there in the 19th century and owned six sugar plantations, the broadcaster reported.
BBC News journalist Laura Trevelyan, who is based in the US, tweeted: “The Trevelyan family is apologising to the people of Grenada for the role our ancestors played in enslavement on the island, and engaging in reparations.”
The family intends to donate £100,000 to establish a community fund for economic development on the island, the BBC said.
Ms Trevelyan said seven family members will travel to Grenada this month to issue a public apology.
The reporter, who visited the island for a documentary, told the BBC the experience had been “really horrific” and that she “felt ashamed” on seeing the plantations where slaves were punished and the instruments of torture used to restrain them.
She said: “You can’t repair the past – but you can acknowledge the pain.”
She said the Trevelyans had received about £34,000 in 1834 for the loss of their “property” on Grenada, which is thought to be the equivalent of about £3 million in today’s money.
She acknowledged that giving £100,000 almost 200 years later could seem “inadequate”, but added: “I hope that we’re setting an example by apologising for what our ancestors did.”
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