Chevron alleges misconduct by Mr. Donziger in its effort to defeat a lawsuit brought by Ecuadorian plaintiffs seeking billions of dollars for the environmental contamination by Chevron’s predecessor in Ecuador, Texaco, and prevail in an international arbitration case. The company claims, and Mr. Donziger denies, that he improperly influenced Ecuadorian officials and experts, and prodded authorities to seek criminal charges against two of the oil company’s attorneys.

Earlier this year, Judge Kaplan ordered the production of outtakes from “Crude,” a documentary about the litigation by filmmaker Joseph Berlinger, and Chevron has used those outtakes to press its case that Mr. Donziger should hand over documents and be deposed.