WASHINGTON—House Republicans threatened to hold Hunter Biden in contempt of Congress if he fails to appear next week for closed-door questioning, escalating an already tense standoff over lawmakers’ demand for testimony from President Biden’s son.

In a letter Wednesday to the younger Biden’s lawyer, two House Republicans insisted that he appear for a closed-door deposition on Dec. 13, the date set by a subpoena issued to him last month. Hunter Biden has offered to testify publicly but resisted sitting for private questioning, with his lawyer Abbe Lowell raising concerns that House Republicans would selectively leak portions of any testimony.