AP
The comments were apparently made off-air while Brzezinski's co-host, Joe Scarborough, was on the phone with Trump's communications director Sean Spicer.
"Let it be an arms race - we will outmatch them at every pass and outlast them all," Brzezinski said Trump told her.
Trump tweeted yesterday that "the United States must greatly strengthen and expand its nuclear capability until such time as the world comes to its senses regarding nukes."
Washington Post columnist David Ignatius told Morning Joe that while Trump's "initial move is to unbalance things, to disrupt, to say the unsayable," his tendency to double down on controversial comments "when he is cornered" is worrisome.
"You can create space, but the space is then there to be able to negotiate, not to throw more wood on the fire," he said. "If you say, 'What I really do want is an arms race,' you're going to lose the country. The country doesn't want another arms race."
Russia on Thursday also called for strengthening "the military potential of strategic nuclear forces" in the country.
Trump's proclamation comes one day after he met with top Pentagon officials, including the Air Force deputy chief of staff for strategic deterrence and nuclear integration, a post that broadly oversees the Air Force's nuclear weapons capabilities.
The Trump transition team did not respond to request for comment.
Watch a portion of the Morning Joe segment below:
BREAKING: Trump to #morningjoe on the nukes tweet: 'Let it be an arms race' https://t.co/X9FMNtgKPK
- Morning Joe (@Morning_Joe) December 23, 2016