Democrats said that memo mischaracterised sensitive information and was meant to discredit Special Counsel Robert Mueller's investigation of potential collusion between Mr Trump's 2016 campaign and Russian Federation.
Schiff disputed the president's claims that Democrats acted in bad faith when crafting their memo, saying "the hypocrisy of this just kind of reaches out and grabs you by the throat".
On Friday evening, the White Houseannounced Trump was not ready to release the Democratic memo. "Clearly the president has something to hide", she said.
But congressional Democrats and Republicans, including House Speaker Paul Ryan of Wisconsin and Rep. Trey Gowdy of SC, who helped draft the GOP memo, have said it shouldn't be used to undermine the special counsel.
"The memorandum contains numerous properly classified and especially sensitive passages", White House Counsel Don McGahn said.
McGahn said Trump was "inclined to declassify" the Democratic memo, and encouraged the committee to make the changes that he said the Justice Department had identified as important for "national security and law enforcement interests". Trump has repeatedly denied any collusion.
Mr Mueller took over the investigation from the FBI.
The dueling memos from different members of the House Intelligence Committee relate to an ongoing investigation by special counsel Robert Mueller of Russian meddling in the 2016 USA presidential election. Rep. Adam Schiff of California, the top Democrat on the House Intelligence Committee, wants to show the Nunes memo is misleading, but also has to walk a fine line by not disclosing intelligence and more about the FISA process - a tightrope walk the White House can exploit. The FBI didn't suggest that there was anything in the GOP memo that was inaccurate.
The president has recently ratcheted up his criticisms of the FBI's handling of probes into Hillary Clinton's email server and has called the House Intelligence Committee's investigations into possible Russian meddling in the 2016 election a witch hunt.
The four-page GOP memo, released last week, argues that Federal Bureau of Investigation surveillance of former Trump campaign official Carter Page was authorized in part because of claims in the unverified Trump dossier authored by former British intelligence agent Christopher Steele. It gets released even though the Department of Justice objected strenuously, in public and in private, calling the memo "extraordinarily reckless".
Christopher Steele, a former British intelligence officer, is the author of the Trump-Russia dossier. The previously released Republican text alleges anti- Trump bias in the investigation of Russian meddling in the 2016 U.S. election that brought Trump to power.
Democrats said that memo "cherry-picked" details. One of them praised a Justice Department official who opposed Trump on immigration.
Kim Jong Un's sister aims to score playing charm game
Pence, who led the United States delegation to the Olympics, has accused Pyongyang of using the event for its own ends. Meanwhile, relations between South Korea and North Korea have shown promising signs of improvement in recent days.