to bribe is correct because grease the palm means to give someone money secretly or dishonestly to get a favor.

This idiom or phrase is useful because it helps students understand the intended meaning beyond the individual words.

Examples: The contractor tried to grease the palm of an official to win the contract. A fair system does not allow anyone to grease the palm of a judge.

This idiom is figurative. It does not mean literally putting oil or grease on a hand or machine.