The Justice Department is seeking to question former Vice President Mike Pence as a witness in connection with its criminal investigation into former President Donald J. Trump’s efforts to stay in power after he lost the 2020 election, according to two people familiar with the matter.

Mr. Pence, according to people familiar with his thinking, is open to considering the request, recognizing that the Justice Department’s criminal investigation is different from the inquiry by the House Jan. 6 committee, whose overtures he has flatly rejected.

Complicating the situation is whether Mr. Trump would try to invoke executive privilege to stop him or limit his testimony, a step that he has taken with limited success so far with other former officials.