The Today programme presenter told the former Ukip leader: “You’ve run seven times and lost seven times.”
That prompted Farage to hit back: “I’m sick to death of your condescending tone.”
The pair clashed following the resignation of NatWest chief executive Dame Alison Rose in the wake of the row over Coutts bank’s decision to cancel Farage’s account with them.
Farage had alleged the decision was “political” after he obtained an internal document that said it was partly because his views were not “aligned” with the bank’s.
The 40-page dossier refers to the ex-MEP as “xenophobic and racist” and a former “fascist”.
That called into question a report by BBC business editor Simon Jack, who had claimed that Farage lost his account because he lacked the funds needed to hold it.
Rose last night admitted she had spoken to Jack about Farage’s Coutts account and in the early hours of this morning it was confirmed that she had resigned.
During Farage’s Today programme interview, Robinson told him: “You know pretty much whatever you do there are people saying ‘I know what this is about, he wants to get back into again’.
“I know you’ve run seven times and lost seven times ...”
But Farage interjected: “I’m really not going to have this. I’m sick to death of your condescending tone.”
Robinson replied: “I was teasing you.”
Farage hit back: “Actually you weren’t. What you should say to people is ‘you’re the only person in British history who’s won two national elections leading two different parties’.”
Undeterred, Robinson said: “Are you coming back then, given your previous success?”
Farage responded: “No, I’m now a champion and a crusader for men and women who’ve been closed down by the banks and I want cultural changes within the banks and I want legal changes.”
Robinson said: “I was teasing you. You and I have had conversations about your parliamentary runs in the past when I’ve done interviews.
“You’re not about to make another run before next year’s election?”
Farage said: “I’m not about to do that, so.”
Treasury minister Andrew Griffith will today meet with the heads of 19 banks to tell them customers should not have their accounts closed because of their political views.
Welcoming Rose’s resignation, he tweeted: “It is right that the NatWest CEO has resigned. This would never have happened if NatWest had not taken it upon itself to withdraw a bank account due to someone’s lawful political views. That was and is always unacceptable.
“I hope the whole financial sector learns from this incident. Its role is to serve customers well and fairly - not to tell them how or what to think.”