WASHINGTON — The White House put some reconstructed transcripts of delicate calls between President Trump and foreign officials, including President Vladimir V. Putin and the Saudi royal family, into a highly classified computer system after embarrassing leaks of his conversations, according to current and former officials.The handling of Mr. Trump’s calls with world leaders has come under scrutiny after questions over whether a transcript of a July 25 call with the Ukrainian president, Volodymyr Zelensky, was improperly placed into this computer system.The latest revelations show the focus that White House officials put on safeguarding not only classified information but also delicate calls with Mr. Trump, the details of which the administration did not want leaked.A whistle-blower complaint accuses officials of trying to “lock down” access to information about the conversation with Mr. Zelensky by improperly storing the rough transcript of the July 25 call in the highly classified system after the call took place. In the case of the calls with the Saudi royal family, the restrictions were set beforehand, and the number of people allowed to listen was sharply restricted. The Saudi calls placed in the restricted system were with King Salman, Crown Prince Mohammed bin Salman and Prince Khalid bin Salman, who at the time was the Saudi ambassador to the United States. While the calls included delicate information about Mr. Trump’s discussions about the killing of the journalist Jamal Khashoggi, there was no apparent evidence of impropriety by Mr. Trump, said a person familiar with the matter.The access restrictions placed on the calls with Mr. Putin of Russia and the Saudi royal family were first reported Friday night by CNN.The practice began after details of Mr. Trump’s Oval Office discussion with the Russian foreign minister, Sergey V. Lavrov, leaked to the news media, leading to questions of whether the president had released classified information, according to multiple current and former officials. The White House was particularly upset when the news media reported that Mr. Trump had called James B. Comey, the former F.B.I. director, a “nut job” during that same meeting, according to current and former officials.The White House had begun restricting access to information after initial leaks of Mr. Trump’s calls with the leaders of Mexico and Australia. But the conversation with Mr. Lavrov and Sergey I. Kislyak, then the Russian ambassador to the United States, prompted tighter restrictions.Several current and former officials played down the significance of placing the classified calls into the secure system, saying it made sense to restrict the calls given the number of leaks from the Trump White House.Nevertheless, the use of the system has come under scrutiny after the unclassified version of the whistle-blower complaint was made public. The complaint raised questions that the July 25 call with the Ukrainian president had been improperly placed in the classified system, suggesting that officials put the reconstructed transcript into a system meant to protect the nation’s most sensitive secrets.The Trump administration said on Friday that National Security Council lawyers had made the decision to place the rough transcript of that phone call into a highly classified computer system accessible to only a small number of officials.“N.S.C. lawyers directed that the classified document be handled appropriately,” said a senior administration official. The statement was also first reported by CNN. But the official did not actually say how the document was handled, nor address the whistle-blower’s specific charge that the reconstructed transcript, in what would be a highly unusual action, was moved from a computer system widely accessible to National Security Council officials to one reserved for those with code-word clearance to handle the country’s most closely guarded secrets like covert operations and foreign surveillance.A White House spokesman did not respond when asked about that specific claim. Democrats in Congress and former N.S.C. officials and lawyers in both parties have said such an action, if motivated by a desire to conceal Mr. Trump’s efforts to put political pressure on the Ukrainian leader, would be far from appropriate, and at a minimum, unethical. But in a combative exchange with reporters later in the day, Kellyanne Conway, a White House counselor, repeated the spokesman’s language, saying that “as I understand, the document was handled appropriately at all times.”“I think the most important thing about said document is that the whole world has access to it now,” Ms. Conway said, citing its release by the White House this week.Democrats say the president abused his power by conditioning aid for Ukraine on whether its government investigated one of his 2020 campaign rivals, former Vice President Joseph R. Biden Jr., and his son Hunter Biden, and will examine whether that constitutes a high crime and misdemeanor. The treatment of the document will form part of that inquiry. In his complaint, the whistle-blower said that unnamed White House officials told him that they had been “‘directed’ by White House lawyers” to remove the record of the call from the National Security Council’s main computer system and load it into one managed by the agency’s intelligence directorate that is not connected to the main system and that requires special permissions and enhanced security clearances to access. That would have the effect of vastly reducing the number of people who can read — and therefore leak — the document, in what the whistle-blower described in his complaint as an acknowledgment that the president’s comments to Mr. Zelensky had been highly improper. “One White House official described this act as an abuse of this electronic system because the call did not contain anything remotely sensitive from a national security perspective,” the whistle-blower wrote. His complaint also alleged that other, unspecified presidential transcripts had received similar treatment.The administration official did not name any of the lawyers involved. The National Security Council is part of the White House and advised by lawyers who report to the White House counsel, Pat A. Cipollone. The National Security Council’s chief legal adviser is John Eisenberg, a Cipollone deputy. Mr. Eisenberg also had a role in conducting a preliminary inquiry into the allegations, well before the whistle-blower complaint was filed. The C.I.A. officer who would become the future whistle-blower first contacted, anonymously, the C.I.A.’s general counsel, Courtney Simmons Elwood, days after the July 25 call. Ms. Elwood then contacted Mr. Eisenberg to begin his inquiry. Mr. Eisenberg and his team began calling people in the N.S.C. about their concerns about the call, according to people familiar with the matter. It is not clear if that inquiry included questions of how the records were handled.Ms. Conway pleaded ignorance of the details of how the N.S.C. handles records of foreign leader calls, saying that “the people who handle such things said it was handled appropriately.” After transcripts of conversations Mr. Trump had with the leaders of Australia and Mexico leaked into the media early in his presidency, she said, “my understanding is that we changed some of the protocols” regarding how records of such calls are managed. Those changes have included limiting the distribution of the call transcripts, something that had been known for some time. Ms. Conway dodged specific questions about whether it would have been improper for the White House to place the N.S.C.’s reconstructed transcript of the Ukraine call into the more secure computer system. Mr. Trump continued to rage on Twitter on Friday over criticism of his phone call with Mr. Zelensky while he and his allies maintained their effort to deflect attention from the president’s actions and onto the former vice president and Hunter Biden. “If that perfect phone call with the President of Ukraine isn’t considered appropriate, then no future President can EVER again speak to another foreign leader!” Mr. Trump tweeted.