Former Starbucks CEO Howard Schultz said Monday that he doesn't plan to respond to remarks President Trump made on Twitter saying Schultz doesn't have the "guts" to run for president.

Schultz, during an interview with CNBC's Andrew Ross Sorkin at a Barnes & Noble in New York, called Trump's tweet "childish."

"I'm not going to respond to that. It's childish. I'm not trying to win the Twitter primary," Schultz said.

Trump sent the tweet Monday morning after Schultz said Sunday during an interview with "60 Minutes" that he is seriously considering running for the president as an independent in 2020.

"Howard Schultz doesn't have the 'guts' to run for President! Watched him on @60Minutes last night and I agree with him that he is not the 'smartest person.' Besides, America already has that! I only hope that Starbucks is still paying me their rent in Trump Tower!" Trump tweeted.

Schultz blasted Trump during the "60 Minutes" interview, calling him not qualified to serve as president. He also said that both Democrats and Republicans are "consistently not doing what is necessary on behalf of the American people."

Some Democrats have expressed concern that Schultz's possible independent bid would benefit Trump by taking votes away from the Democratic presidential nominee.

During Schultz's interview with CNBC on Monday, he said he doesn't intend to "do anything" to help Trump remain in office but added that he believes he could win if he runs.

A heckler shouted at him during the event, urging Schultz not to run.

"Don't help elect Trump, you egotistical, billionaire asshole," the person could be heard saying.