Kellyanne Conway, a senior advisor to US President-elect Donald Trump, said "possibly" to MSNBC, when asked if Giuliani was going to be the head of DNI.
Giuliani met Trump early in the morning.
"I'm not going to discuss what I'm interested in or what the President-elect is thinking and mulling over. That's not fair. That's not the right way to do it. All of this has to be done very privately. I don't mean only with regard to me, I mean with regard to everyone else," Giuliani told reporters after the meeting.
"The President-elect, as well as it should be, with input from the Vice president-elect," she told MSNBC.