The handling of the Ukraine call transcript has raised alarms among Democrats in Congress and former national security officials of both parties, including ones who served in the Trump administration. They agreed that nothing about the conversation appears to warrant placement in the extra-secure system.

“We need to look into the allegation that this may not be the only communication of a potentially corrupt character that was shielded by this classified information computer system abused for that purpose,” Representative Adam B. Schiff, Democrat of California and the chairman of the House Intelligence Committee, told reporters on Thursday at the Capitol.

Mr. Schiff and other Democrats say it appears that the White House sought to hide Mr. Trump’s effort to pressure Mr. Zelensky to investigate former Vice President Joseph R. Biden Jr., a possible opponent in the 2020 presidential election, and his son Hunter Biden, as well as to pursue conspiracy theories that hold that Russia did not meddle in the 2016 presidential election.

“During my tenure, I had no knowledge of documents being moved out for political sensitivity,” said a former Trump White House official familiar with the system. “I don’t know the legality of what was done. It certainly feels unethical.” The official said that decisions to move documents to different levels of classification typically involve the National Security Council’s legal office.

All officials at the agency, whose size has ranged from about 100 to 150 staff members in recent years, have security clearances that allow them access to a shared classified computer network on which countless documents are stored, many of them categorized at relatively low levels of secrecy.

Stored on the less-classified network are transcripts of presidential calls with foreign leaders, which past White Houses have actively distributed widely among national security and foreign policy officials on secure email systems.

After embarrassing leaks in 2017, in which two transcripts of Mr. Trump’s calls with foreign leaders were published in the news media, the Trump White House cut the number of people to whom phone call records were distributed.