The PA news agency has also reported Croxall has been taken off air until the investigation is complete.
It follows complaints on social media that she displayed bias while hosting Sunday's newspaper review programme.
During her introduction, after Boris Johnson ruled himself out from the Conservative leadership race, Croxall told viewers she felt "gleeful".
The programme aired around 90 minutes after the former prime minister claimed to have enough support to progress in the Tory leadership race, but had decided to withdraw.
As The Papers programme began just after 22:30 on Sunday evening, Croxall told viewers: "Well this is all very exciting isn't it? Welcome to our lookahead to what the papers will be bringing us tomorrow. Am I allowed to be this gleeful? Well I am."
She was then seen asking her guests: "Can we even show you the front pages just yet, have they arrived? No they haven't arrived. It's all a little bit, you know, lastminute.com isn't it? Because all the front pages were probably out of date by the time we received them."
Croxall also suggested during the programme that her comments could have breached BBC guidelines.
Responding to a guest's joke aimed at Mr Johnson, she said: "I shouldn't probably (laugh). I'm probably breaking some terrible due impartiality rule by giggling."
On Monday, a BBC spokesperson said: "BBC News is urgently reviewing last night's edition of The Papers on the News Channel for a potential breach of impartiality.
"It is imperative that we maintain the highest editorial standards. We have processes in place to uphold our standards, and these processes have been activated."
One Conservative MP, Neil O'Brien, called the the presenter's comments "self destructive".
Nick Timothy, previously an adviser to former prime minister Theresa May, added: "Whatever you think of Johnson, if you care about the BBC you cannot think this is in any way acceptable from one of its presenters."
It has not been made clear by Croxall or the BBC whether she was referring to Mr Johnson's withdrawal or to the fast-moving news agenda.
However, she told one viewer on Twitter to consider the context of her comments. She added she was due to return to the airwaves on Monday evening.