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Michelle Brown used derogatory comments about the Labour MP for Streatham, Chuka Umunna, in a call in May 2016 to her then senior adviser Nigel Williams.
Ms Brown said her language was "inappropriate" and has "apologised to anyone that has been offended by it."
UKIP chairman Paul Oakden said the party does condone her personal views.
Ms Brown, who called Mr Umunna a "coconut", was also recorded using an abusive remark about Tristram Hunt, who was then Labour MP for Stoke-on-Trent Central.
Mr Williams, who was her senior adviser for 12 months, was sacked by Ms Brown in May.
Ms Brown said in a statement: "The point I was making is that because of his considerable wealth and privilege, Chuka Umunna cannot possibly understand the difficulties and issues that the average black person faces in this country any more than I can, and I stand by that assertion.
"I do however accept that the language I used in the private conversation was inappropriate and I apologise to anyone that has been offended by it.
"As far as the language I used about Mr Hunt is concerned, it was a private conversation and I was using language that friends and colleagues often do when chatting to each other."
Ms Brown's comments have been referred to the assembly's standards commissioner.
Mr Williams said he believed Ms Brown should resign from her seat and UKIP's national executive committee should remove her from the party.
Mr Oakden said UKIP "obviously does not condone the personal views expressed by Michelle Brown".
"We will conduct an investigation into this matter, the findings of which will be passed through to our National Executive Committee so that they might consider disciplinary action," he said.
"We will also investigate whether a UKIP member and official surreptitiously recorded a private telephone conversation with Michelle Brown and then disseminated it without her consent, more than a year after the event."
The assembly's Labour Group condemned the "absolutely outrageous language" and said "anything less than immediate suspension would be a clear endorsement of Michelle Brown's racist slur."
Plaid Cymru leader Leanne Wood said: "This racism reflects poorly on our parliament - The National Assembly for Wales - and that's why her party should take action on this."
This is not the first controversy Ms Brown has faced - in February, she was forced to deny claims she had smoked "recreational drugs" in a Cardiff Bay hotel room.
Her spokesman said the smell was caused by the AM smoking a strong tobacco product.