Cummings said that when it was under Republican control in March 2017, his committee started investigating whether White House officials were using personal email and messaging accounts to conduct official business.
That, Cummings said, violates the Presidential Records Act.
The revelations are particularly embarrassing for the president, who during the 2016 campaign called for the jailing of Hillary Clinton over her own use of a private email account and server to exchange top secret information.
Cummings said the administration has not provided the panel with a "single piece of paper" since he requested a batch of records - as well as a briefing by White House officials - last December.
Lowell would not tell the committee whether Kushner had ever used WhatsApp to discuss classified information.
A spokesman for Lowell provided Lowell's response to Cummings on Thursday afternoon. Cummings noted that in October 2017, White House lawyers briefed committee staff and "confessed" that several White House employees had failed to forward official records from their personal email accounts to their government accounts within 20 days, although they refused to identify the staffers. He said he also informed the committee that Kushner complies with all protocols involving classified information.
Rep. Elijah Cummings, D-Md., said in a letter that Trump's lawyer, Abbe Lowell, informed the committee late previous year that Trump doesn't preserve official emails she receives in her personal account if she doesn't respond to them.
A key House Democrat is renewing demands that the White House turn over documents about the use of private texts or emails by Jared Kushner, saying Kushner's lawyer acknowledged that the senior aide used the non-secure WhatsApp application to communicate with foreign leaders.
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"I did not specify who they were", Lowell said. The attorney noted Kushner has "numerous friends and contacts overseas".
Lowell says he told Cummings their discussion was in regard to a period before September 2017. Lowell added he later told the committee, "Now she always forwards official business to her White House account".
Previous attempts by the committee to obtain information on the use of private email have been rebuffed by the White House, setting up the latest battle between the Democratic-controlled body and the president. An attorney for McFarland declined to comment.
Cummings added that Lowell also said his other client, Ivanka, sends personals email messages relating to her duties to her official White House email account only when she responds to them.
The Democratic head of a USA congressional investigative panel on Thursday pressed the White House for information on whether President Donald Trump's son-in-law and adviser, Jared Kushner, used the unofficial WhatsApp messaging tool to communicate sensitive or classified information with foreign leaders.
Mr Cummings also wrote that his committee has obtained new information about other White House officials that raises additional security and federal records concerns about the use of private e-mail and messaging applications.
Section 2209 of the statue stipulates White House officials are required to forward a "complete copy" of "official business conducted using non-official electronic messaging accounts" to "an official electronic messaging account of the President, Vice President, or covered employee" no later than 20 days after the creation of the communication.
He said documents were due January 11, and the White House has failed to provide any additional documents for the committee's bipartisan investigation.