WASHINGTON—House Intelligence Committee Chairman Adam Schiff said Democrats would seek a full airing of special counsel Robert Mueller’s coming report on possible collusion between the Trump campaign and Russia, vowing a legal battle over documents and testimony if needed.

Mr. Schiff, of California, said that Attorney General William Barr should make the underlying evidence from Mr. Mueller’s investigation public, even if President Trump isn’t criminally implicated in any wrongdoing. Mr. Trump has repeatedly denied working with Russia in what the U.S. has said was a broad campaign to help him win the 2016 election.

“We will obviously subpoena the report, we will bring Bob Mueller in to testify before Congress, we will take it to court if necessary,” Mr. Schiff said on ABC’s “This Week.”

Mr. Schiff added: “When you take the position that the president cannot be indicted and the only remedy for improper, illegal or other conduct is impeachment, then you cannot withhold that information from Congress, or essentially the president has immunity. So that cannot be allowed to be the case.”

During Mr. Barr’s recent confirmation hearing, Democrats sought assurances that he wouldn’t try to impede Mr. Mueller’s probe, particularly in light of a June 2018 memo that Mr. Barr wrote that was strongly critical of aspects of the Mueller inquiry. Mr. Barr vowed to bring as much transparency as possible to Mr. Mueller’s findings, though he wouldn’t promise to release the special counsel’s final report.