Kelly left open the door that something improper could have happened, and said he hopes investigations by Justice Department special counsel Robert Mueller and congressional committees will uncover it if that is the case.
Kelly was testifying before the Senate Homeland Security and Governmental Affairs Committee, where he was grilled by Montana Democratic Sen. Jon Tester over Kelly's comments
last month that "any channel of communication, back or otherwise, with a country like Russia is a good thing,"
At the time, Kelly was responding to a report
that Kushner may have discussed setting up a secret channel for communications with Russia's ambassador during the presidential transition.
Tester pushed back on Kelly saying then that it wasn't a big deal.
"I think this is a big deal, because we're talking about Russia," Tester said, noting that he and Kelly are old enough to remember drills during the height of the Cold War that sent children hiding under their desks.
Kelly said plenty of back-channel communications happen among government officials that are not necessarily nefarious, and he has had his own back-channel communications with foreign leaders.
"We have to make the assumption that Jared Kushner is a great American," Kelly said, saying he believes Kushner had a security clearance at the time.
"Now, if he was trying to open back channel communications to pass information through that back channel ... if it's official, it's a whole different dynamic," Kelly added.
Tester asked, though, if it would be normal to conduct such communications at the embassy of a non-friendly country like Russia, and Kelly said he wasn't sure.
The senator pushed Kelly to get to the bottom of the situation, saying that he has the "credentials and respect ... to find out what the hell is going on."
"I appreciate your faith in the system, I'm going to tell you, whether classified information was delivered or not, I find this unacceptable," Tester said. "I just do. To have somebody who is the son in law of the President who would go to Russia, the country I was told to hide under the desk from. ... There's all sorts of stuff going on here, and as Claire (McCaskill), the ranking member said, there's so much going on here that we don't know which direction the investigation needs to happen."