Trump told reporters at the White House on Thursdaythat he discussed the idea with Pompeo and "he (Pompeo) likes the idea of having somebody in there with him and I do too", reports Xinhua news agency.
Trump added that he has 15 candidates for the job, but did not offer further details.
It was reported on Wednesday that the Trump administration officials had been discussing the possibility of replacing Bolton with Pompeo, which means that America's top diplomat would assume the position of the NSA and do both jobs.
On Tuesday, Trump fired Bolton with whom the President said he "disagreed strongly".
Trump is expected to announce Bolton's successor next week, with Deputy NSA Charles Kupperman serving as an acting head.
US media has put together a list of possible candidates who are likely to be tapped for the job, including Kupperman, Special Representative for the Democratic People's Republic of Korea Stephen Biegun, Special Representative for Iran Brian Hook, Douglas Macgregor, a Fox News commentator, and US Ambassador to Germany Richard Grenell.