The president's sons, his son-in-law Jared Kushner, The FBI, and certain members of the Trump organization and Wiki-leaks are among those who received requests from the panel.
"I think there are enough American people that don't even like Trump that will say this is too much", Huntsman insisted.
The White House said its lawyers and "relevant White House officials" would review the Judiciary Committee's demand for documents and "respond at the appropriate time".
The probe comes following mid-term elections in the U.S. that gave Democrats a majority over Trump's Republicans in the House.
Cohen told Congress in testimony last week that he was cooperating with Manhattan federal prosecutors on several ongoing probes and flagged other potential illegal activity by Trump.
The chairmen of three other House panels also joined the fray, demanding the White House produce details of communications between Mr Trump and Russian President Vladimir Putin.
Probe into whether Donald Trump committed abuse of power
Despite Nadler's hesitation on impeachment, later in the show, House Minority Leader Rep. Eric Swalwell of California, said they would learn a lot more over the next few weeks.
In a letter to House Oversight and Reform Chairman Elijah E. Cummings (D-Md.), White House Counsel Pat A. Cipollone said the committee request for the information was "without legal support, clearly premature, and suggests a breach of the constitutionally required accommodation process".
Cobb also disagreed with the president's view that the Russian Federation probe is politically-motivated "witch hunt".
Cobb was notorious during his White House days for predicting the Mueller investigation was near an end - then revising his prediction for a new deadline.
The White House is refusing to hand over to Congress documents regarding its process for security clearances, possibly paving the way for a second round of subpoenas to hit the Trump Administration.
Nadler said the document requests, with responses to most due by March 18, are a way to "begin building the public record". With so many investigators encircling him, it would be shocking if Trump weren't hoping to find a way out.
American intelligence agencies and Mueller have said Russian Federation was the source of hacked material released by the anti-secrecy group WikiLeaks during the campaign that was damaging to Democrat Hillary Clinton's presidential effort.
The top Republican on the committee, Georgia Rep. Doug Collins, said Nadler was "recklessly prejudging the president for obstruction" and pursuing evidence to back up his conclusion.
As the Judiciary Committee's voluminous requests circulated around Washington on Monday, the president's outside array of former allies, associates and staffers communicated among themselves about who was named in the requests and whether they faced new legal jeopardy.