The special counsel probing former President Donald Trump’s handling of classified material wants one of the ex-president’s lawyers to testify before a grand jury under the theory some of their conversations abetted a potential crime.

The effort, first reported by The New York Times, signals an aggressive push by special counsel Jack Smith to find a way around attorney-client privilege that would normally shield conversations between Mr. Trump and his lawyer, Evan Corcoran.

Mr. Smith wants to invoke the crime-fraud exception as he probes whether Mr. Trump and his team obstructed justice by failing to return certain documents to the National Archives.

The unfolding matter pivots on statements Mr. Corcoran made in response to subpoenas for the record last year, before a raid on Mr. Trump’s Mar-a-Lago home.

Mr. Corcoran recently appeared before a grand jury in Washington and asserted attorney-client privilege, according to The Times’ report.

A Trump spokesman characterized the latest effort as a political witch hunt designed to upend Mr. Trump’s 2024 bid.

Mr. Smith is investigating the classified document affair and Mr. Trump’s efforts to question the 2020 election outcome before a pro-Trump mob attacked the U.S. Capitol on Jan. 6, 2021, in an effort to stop the certification of the results.

The special counsel recently subpoenaed former Vice President Mike Pence to compel information about efforts to invalidate the formal results.

Mr. Pence plans to contest the subpoena by invoking the Speech and Debate clause of the Constitution that protects legislative activity, given his role as president of the Senate.