Trish Adudu was walking to her car in Coventry city centre when a man cycled past and made the remark.
The BBC Coventry and Warwickshire presenter said she witnessed the same person abusing an Asian man moments earlier.
Police have vowed to find the perpetrator of the "sickening" attack.
It comes after the West Midlands force's Chief Constable Dave Thompson promised a "zero-tolerance" stance on hate crime following the EU referendum result.
Several racial attacks have been reported in the wake of the decision to leave the EU.
Trish told listeners the attack happened as she was getting into her car at around 09:30 BST on Wednesday.
She said: "He cycled over to what looked like an Asian student and was basically saying 'Get out of here. Go back home. Haven't you heard the vote?'
"And then he cycles around, sees me looking at him in shock, because it was so loud and so angry.
"And then he says to me 'yeah, that goes for you as well'. He starts calling me the N word and told he to 'go home'."
Insp Alastair Orencas appealed for witnesses who saw the abuse to come forward.
"There is absolutely no place for this type of sickening and deplorable behaviour on the streets of our city and we will be doing everything we can to identify the perpetrator," he said.
"I would also appeal directly to this individual - if it was you who shouted these words, do the right thing and hand yourself in."
Trish added: "This type of vile abuse needs to stop and we need to work together to try to get rid of it.
"I would love for the other man who was there to come forward, because this is not just about me.
"If the guy on the bike could be so vile in such a short period of time, then what else could he go on to do? Who's next? Someone is going to get hurt."
A police investigation has been launched after a BBC radio presenter was racially abused in the street.