This memo was written by Democrats on the panel who are pushing back against a GOP document, declassified by Trump last week, that criticizes the methods the FBI used to obtain a surveillance warrant on a onetime Trump campaign associate.
The Democratic document attempts to counter some of the arguments and evidence put forward by the Republicans.
It also takes attention from the separate investigations by special counsel Robert Mueller and the Senate intelligence committee.
Trump said over the weekend that the GOP memo "totally vindicates" him. Both Republicans and Democrats disputed that, and Democrats also bemoaned the release of formerly classified information and the possibility the precedent could compromise future investigations.
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After the House committee's Monday evening vote, which was unanimous, Rep. Adam Schiff of California, the panel's top Democrat, said he believed the Democratic document would "help inform the public of the many distortions and inaccuracies in the majority memo."
The president now has five days to decide whether to allow the material's publication.
Schiff said he would compare any deletions the FBI and the Department of Justice might request with any White House edits to try to identify any attempts to withhold information for political purposes.
Texas Republican Rep. Mike Conaway, a leader of the panel's Russia probe, said after the vote that parts of the document should not be released.
"There are things in the memo that I would be uncomfortable with if the White House did not redact," he said.
Trump tweeted that Schiff is "one of the biggest liars and leakers in Washington" and "must be stopped."
Schiff quickly shot back: "Instead of tweeting false smears, the American people would appreciate it if you turned off the TV and helped solve the funding crisis, protected Dreamers or ... really anything else."
White House spokesman Raj Shah said merely that consideration of a release would "allow for a legal review, national security review led by the White House counsel's office.