Following Donald Trump's 37-count indictment, a number of prominent Republicans, including potential presidential candidates Mike Pence and Asa Hutchinson, denounced the handling of secret material by the former US president on Sunday.

The remarks were made on the first round of significant weekend political talk shows after Trump pleaded not guilty in a Miami courthouse on Tuesday to mishandling some of the most delicate secrets of the US government. They included severe criticism from the former Trump Defense Secretary Mark Esper.

Read: Trump says has right to take 'whatever' documents home

These were among the several documents that Trump carried with him from the White House in an effort to hide them from investigators.

The comments made on Sunday are a stark contrast to many Republicans in Congress who have either defended Trump or refrained from criticizing him.

Pence, Trump's former Vice President, said on NBC's Meet the Press, in reference to his ex-boss' actions in the documents scandal, "I can't defend what is alleged."

Former Arkansas governor William Asa Hutchinson went further, describing the accusations as "serious and disqualifying."

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"I think that he should drop out" of the 2024 race, he told ABC's This Week.

Read: DeSantis, Pence defend Trump after indictment

Trump is accused of compromising national security by illegally storing top secret military plans and nuclear weapons information at his Mar-a-Lago resort in Florida. He has claimed that the Department of Justice is being used against him.

The case is one among many legal issues that could prevent him from winning re-election to the White House in 2024.

"If the allegations are true, that it contained information about our nation's security... it could be quite harmful to the nation," Esper told CNN, adding that "no one is above the law" and calling the revelations "disturbing."

However, other Republican candidates for president are in the difficult position of attempting to differentiate themselves from Trump, the current front-runner for the party's nomination, without losing the former President's devoted and significant support base.

"The former president deserves his day in court," Pence added. "I want to reserve judgment about this until he's had an opportunity to take his case into the courtroom."

"I don't know why some of my competitors in the Republican primary presume the president will be found guilty."