Trump, at a joint news conference with visiting Australian Prime Minister Malcolm Turnbull, said: "General Kelly respects Jared a lot and General Kelly will make that call" on how to resolve the issue.
"I will let General Kelly make that decision".
Problems with security clearance for White House aides surfaced this month in fallout over the abrupt resignation of White House staff secretary Rob Porter over allegations of domestic abuse against two former wives, reports Reuters.
A White House spokesman declined to comment on the status of Kushner's clearance or on information relayed by Rosenstein to McGahn.
As someone who meets regularly with foreign officials and reads classified intelligence, Kushner would typically have a fast-tracked background investigation, security-clearance experts said.
"Nothing that has taken place will affect the valuable work that Jared is doing", White House Press Secretary Sarah Huckabee Sanders told reporters earlier this week.
Kelly set Friday as the deadline for adjudicating the cases of Kushner and others in clearance limbo. "He continues and will continue to be a valued member of the team".
Kelly said last week that officials who applied for security clearances before July, but still have not been granted permanent access, would have their temporary clearances taken away. "He'll continue to do the important work that he's been focused on for the previous year".
Their wait-and-see mode ended abruptly last week, when Kelly issued a new policy that would block staff with interim clearances from receiving top-secret information as of Friday.
In response to a question about whether he'd grant Kushner a waiver so he could keep working on sensitive foreign policy issues without a security clearance, Trump said his son-in-law was a "high-quality person" who had been treated "very unfairly". "There is no truth to any suggestion otherwise". Two people said the deputy attorney general told McGahn the Justice Department had obtained important new information, suggesting it could be an obstacle to his clearance process.
Trump said the US effort to make a deal between the Israelis and the Palestinians is "actually making great headway". "She said, 'Dad, i want to go to Washington".
The source said there were frustrations on both sides between Kushner and Kelly and they wanted to find a solution. Kushner has also come under scrutiny in special counsel Robert Mueller's ongoing investigation into Russian interference in the 2016 election.