Tuesday's fresh criticism comes after Trump's campaign released its May financials, pointing to weaker fundraising figures than would typically be expected at this point in the race.
A spokeswoman for Trump did not immediately return a request for comment.
Trump's campaign committee received only $5.6 million in May, including another loan from the candidate himself, according to a Federal Election Commission filing. It reported disbursements of $6.7 million, ending the month with only $1.3 million on hand.
In a Tuesday morning statement, the Trump camp explained that low figure as the consequence of only really beginning its fundraising efforts at the end of May, and actually described its results thus far as "incredible."
"If need be, there could be unlimited 'cash on hand' as I would put up my own money, as I have already done through the primaries, spending over $50 million dollars. Our campaign is leaner and more efficient, like our government should be," Trump said in the release.
As CNBC reported after the original Cuban tweet storm, Trump has claimed repeatedly that because of his wealth, he doesn't have to make promises to wealthy donors as traditional politicians do.
Trump made that claim while appealing to supporters of Sen. Bernie Sanders, who may be looking for a general election candidate now that Hillary Clinton has become the apparent Democratic presidential nominee.
"I beat a rigged system by winning with overwhelming support — the only way you could've done it," Trump said at the time. "We can't fix a rigged system by relying ... on the very people who rigged it."
Cuban has previously criticized Trump, questioning the real estate developer's claims that he's a billionaire, and saying he doesn't think Trump is "a reflection of a CEO or a reflection of an entrepreneur."
Disclosure: CNBC owns the exclusive off-network cable rights to "Shark Tank," which features Mark Cuban as a judge.
— CNBC's Jacob Pramuk contributed to this report.