Following the incident, Marney apologized for her comments and said that they were taken out of context.
In the series of alleged vile slurs, Henry Bolton's 25-year-old lover Jo Marney said Ms Markle's "seed" would "taint the Royal Family".
The party's ruling National Executive Committee is set to discuss Mr Bolton's private life at a special meeting this week.
Markle, who will Wednesday Harry in May, has spoken publicly and written about being biracial, including how it affected her acting career.
Last year, Harry criticized some media reports and online comments for drawing negative attention to Markle's African-American heritage.
"Also it would be hypocritical to allow her to stay. because he said in the leadership election he was the candidate against racism and nazis so she has to go or he and @UKIP are doomed if we let this behaviour happen in the party", he added.
The ex-model - who has remained in touch with the friend she texted since starting her relationship with Mr Bolton - tells him that she would not have sex "with a negro".
Peter Whittle, former deputy leader of Ukip, called for Marney to be permanently suspended, tweeting: "These are disgraceful remarks".
The Mail on Sunday claimed she made the remarks in a text message to a friend.
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One of the party's MEPs said Mr Bolton, 54, should step aside "quickly", calling him a "naive political lightweight" whose personal life was distracting from policy.
It piles more pressure on Mr Bolton, who has already attracted negative publicity by announcing he was leaving his second wife Tatiana Smurova for Ms Marney.
The pair's relationship became public earlier this month after the Ukip leader confirmed that he had changed his "relationship status" in recent weeks.
The newspaper also published an apology from Ms Marney.
She added: "The opinions I expressed were deliberately exaggerated in order to make a point and have, to an extent, been taken out of context".
During the exchange, Ms Marney - who is from Kent and describes herself as an actor and music journalist - also complains that there is "an immigration problem", saying: "This is Britain".
In replying, Mr Bolton tweeted: "Jo was suspended immediately upon us receiving this information".
"I woke up and saw a missile warning and thought 'no way.' I thought 'No, this is not happening today, '" Malapit said.
Mr Oakden declined to give his own views on Mr Bolton's situation, but said the party needs to be "behind our leader 100% in taking that battle (for Brexit) forward".