Some of President Donald Trump's own staff are questioning whether they want to remain at the White House, Axios Executive Editor Mike Allen told CNBC on Thursday.

"People are watching what he's saying, watching what he's tweeting. We're getting increasing texts, emails, calls from people inside wondering if this is something that they want to remain a part of," he said in an interview with "Power Lunch."

Rumors circulated Thursday morning that National Economic Council Director Gary Cohn might resign, sending stocks lower. However, the White House strongly denied those rumors, calling them "100% false."

Trump has been under fire for his reaction to the deadly white supremacist rally in Charlottesville, Virginia, over the weekend. On Tuesday, he blamed "both sides" for the violence and claimed some of the people participating in the rally were only against the removal of a statue of Confederate Gen. Robert E. Lee.

Then on Thursday he took to Twitter to speak out against removing Confederate statues.

The bipartisan criticism of his remarks has been swift, and on Wednesday, top business executives disbanded an advisory council to Trump.

Axios reported that the president has "left himself on an island inhabited by a shrinking band of true-believer voters."

"The problem for this president is that the exact market forces that he would need to rise, that he would need to get his footing, recover are all people that he's alienated," Allen told CNBC.

The White House did not immediately respond to a request for comment.

— CNBC's Jacob Pramuk and Evelyn Cheng contributed to this report.