The 18-year-old, who is on loan from Chelsea, alleged he was abused while receiving treatment for an injury in the opening minutes of the 2-2 Sky Bet Championship draw at The Den and complained to Forest assistant coach David Kelly at half-time.
Forest on Wednesday published a statement on their official website stressing both clubs were communicating in a bid to "uncover the truth".
It read: "Nottingham Forest treat any incident of this nature extremely seriously and will be liaising closely with the relevant authorities in their investigation of this issue.
"Both clubs are in communication regarding the alleged incident and hope to uncover the truth as quickly as possible for all concerned parties."
A Football Association spokesman told Press Association Sport the governing body will open a full investigation into the alleged incident.
Millwall, who revealed a tunnel steward had been informed of the alleged incident by a member of Forest's coaching staff, believed to be Kelly, claimed Chalobah had not wished to make an official complaint at the time "in the absence of any corroborative evidence".
A statement published on millwallfc.co.uk read: "Millwall Football Club were informed late on Wednesday afternoon by Nottingham Forest Football Club that their on-loan midfielder Nathaniel Chalobah alleges he was subjected to racial abuse during the Sky Bet Championship clash between the two clubs at The Den on Tuesday night.
"It had been reported by a member of the Forest coaching staff to a Millwall tunnel steward at half-time during the match that the player felt he had been racially abused.
"Millwall's stadium manager duly instructed the trackside stewards' supervisor to speak to all stewards in the area where the alleged abuse was heard.
"In addition, five additional trackside stewards were deployed to the area for the second half.
"The Forest coach informed the tunnel steward that the player did not wish to make an official complaint, and in the absence of any corroborative evidence, police were not called and there was nothing reported post-match by the officials, police or Nottingham Forest staff.
"All details of the incident and initial investigation, including names of those involved, were logged in accordance with the protocol for such eventualities.
"In the light of the player's subsequent decision to raise the issue once more, Millwall Football Club will assist and support any police investigation that may ensue, and if evidence of racism comes to light, do everything possible to ensure the perpetrators are brought to justice and receive due penalty.
"However, we are disappointed that the chances of substantiating the allegations have been significantly reduced due to the fact they were not deemed serious enough for an official complaint, which would have led to immediate police involvement, to be made at the time."
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