They reportedly felt that, amid a rapid turnover of senior White House staff, Kushner would continue to have influence with the President.
Kushner reported the possibility of a diplomatic back channel to Mike Pompeo, who was then the Director of the CIA, the New York Times reported.
It might have been expected that Kushner would tell the US State Department instead, but he did not have a good relationship with Rex Tillerson, then the Secretary of State.
Pompeo took the lead in negotiations with North Koreans and subsequently replaced Tillerson as Secretary of State.
The White House and the CIA declined to comment on Kushner's role.
Schulze told the New York Times: "I do not discuss the nature of my business or personal relationships."
Kushner was also said to have been involved in the setting up a diplomatic back channel between Trump's Administration and China early in the presidency.
Kushner reportedly set up a diplomatic back channel through Cui Tiankai, China's Ambassador to Washington, which helped lead to Xi Jinping, the Chinese President, visiting Trump at Mar-a-Lago, his Florida retreat in April last year.
Meanwhile, it emerged that a former confidant of Trump met a Russian individual who offered "dirt" on Hillary Clinton for US$2 million.
Roger Stone, the confidant, told the Washington Post he rejected the offer in May 2016, telling the Russian: "You don't understand Donald Trump. He doesn't pay for anything."
It also emerged that the Russian had previously been an FBI informant. Stone, who met the man at a restaurant in Florida, said he appeared to have been the target of an "FBI sting".