Mr Kushner, a senior adviser to the president, had been receiving top-level security briefings.
However, background checks into Mr Kushner had still not been completed, so he had interim clearance only.
He and other White House aides who have yet to receive permanent clearance will not now get top-level briefings.
Mr Kushner, who is married to Mr Trump’s daughter Ivanka, had access to the President’s Daily Brief, a secret intelligence report.
The moves were confirmed by Mr Kushner’s lawyer to Politico, and by two unnamed officials to the Reuters news agency.
Politico reported that the 37-year-old was informed of the decision on Friday.
Politico quoted Mr Kushner’s lawyer Abbe Lowell as saying that it would “not affect Mr Kushner’s ability to continue to do the very important work he has been assigned by the president”.
A spokesperson for Mr Kushner told the BBC in a statement: “Those involved in the process again have confirmed that there are dozens of people at Mr Kushner’s level whose process is delayed, that it is not uncommon for these clearance reviews to take this long in a new administration, and that the current backlogs are now being addressed.”
It comes as the White House moves to impose greater discipline on access to secrets.
General John Kelly, Mr Trump’s chief of staff, said earlier this month that he would be limiting the number of people with top-level security clearance. It emerged that the president’s former staff secretary, Rob Porter, had been able to work with interim security clearance despite allegations of domestic abuse.
The news of the change in Mr Kushner’s status followed the announcement hours earlier that his spokesman Josh Raffel was to leave the White House.
Meanwhile, the Washington Post reported that Mr Kushner’s contacts with certain foreign government officials have raised concerns with the White House.
The newspaper, citing current and former US officials familiar with intelligence reports on the matter, reported that officials in the United Arab Emirates, China, Israel and Mexico have privately discussed ways in which they could take advantage of Mr Kushner and his business arrangements.
Mr Kushner was given a wide brief in his role as a White House adviser despite a lack of political experience.
He was tasked with brokering a Middle East peace deal and liaising with Mexico.
But he had faced problems with his background checks as he sought to obtain permanent security clearance.
The wealthy New York real estate developer had to refile the national security questionnaire required of all prospective White House employees after making a number of omissions.
Last October, the head of the National Background Investigations Bureau told Congress he has “never seen that level of mistakes” on any security clearance application. (Courtesy BBC)