WASHINGTON—President Trump signaled Monday he would support the public release of special counsel Robert Mueller’s full report on Russian interference in the 2016 election, which he says exonerates him, while Democrats contested that interpretation and pressed for more information on its findings.

Neither Congress nor the White House has received the full report. According to a letter Sunday from Attorney General William Barr to senior lawmakers summarizing the report, Mr. Mueller didn’t find that Mr. Trump and his campaign...