Telfer was forced to interrupt the caller and said that Veitch's earlier comments were "mischievous".
"It was a private conversation ... I said: 'Why is it you want to restrict so many people from competing in sport? I mean, soon you'll be saying thin people can't compete. Fat people can't compete. Short people can't compete. Tall people can't compete. Black people can't compete.'
"And Tony, unfortunately ran into the studio and blabbed that out on air which made me look like a racist," he said on-air.
Telfer said Veitch had apologised to him.
"I took offence [over] what he said because it was - he didn't mean to make me sound racist, but ... at the time I wasn't in the studio with him to point out he what he was saying was misguided and mischievous."
Telfer last night told the Herald the two had later worked out their differences but would discuss the disagreement at the end of Veitch's show this morning.
"He accepts that he could have perhaps chosen his words a little more wisely. And I in turn said some things to him, which in hindsight I probably regret - I got a little heated as he came out of the studio.
"It was a bit of a storm in a teacup and we've both moved on," Telfer said.
Veitch resumed radio work earlier this year following the fallout from his court case involving former partner Kristin Dunne-Powell.
He plead guilty in court to a charge of injuring with reckless disregard after admitting he had kicked Ms Dunne-Powell in the back, while she was on the ground.
In 2005, Veitch apologised for "unjustified" comments after he compared tennis star Serena Williams to an ape.