Gregg also voiced fears that Warren would pursue "a social justice agenda" in her new role at Consumer Financial Protection Bureau.
"My concern is that she would use the agency for the purposes of promoting social justice versus for the purposes of promoting better credit and having a stronger financial system," he said.
"If we don't have banks that are viable and strong, where are we going to be then. She doesn't take this into consideration," Shelby said. "I think you have to have a balance. I think you have to have someone with more objectivity. I think this is a big mistake."
But Republicans are not alone in taking issue with the White House decision to circumvent the Senate process. Democratic Sen. Jeff Merkley of Oregon said he hoped Obama would nominate Warren to a permanent position in charge of the consumer agency not just a special advisory role.
"She is more than deserving of the job and the Senate should have the opportunity to confirm one of the nation's strongest consumer advocates," Merkley said.
The president on Friday did not respond to questions about why he decided not to appoint Warren to a permanent position with the agency.
Not only has Warren's selection upset various Republicans on Capitol Hill but also infuriated the financial industry.
"This may be a calculated move to help fire up some groups ahead of the midterm elections, but it undermines the credibility and effectiveness of this already politicized new agency from day one," said David Hirschmann of the U.S. Chamber of Commerce.
An industry executive, speaking on the condition of anonymity, told ABC News, "She has no experience of running a government agency or working in the financial world. The ideal candidate would have both."
White House officials have said that Warren will "play the lead role" in setting up the agency. In addition, she will "advise the president on policies and programs that are designed to protect the financial interests of middle-class families."
Her title will be Assistant to the President and Special Adviser to the Secretary of the Treasury, heading the Consumer Financial Protection Bureau.
In a blog posted earlier today on the White House website, Warren vowed that the agency -- created by the Wall Street overhaul bill that passed Congress this summer in the wake of the 2008 economic meltdown -- would become "a tough cop on the beat."
"The new law creates a chance to put a tough cop on the beat and provide real accountability and oversight of the consumer credit market," she wrote. "The time for hiding tricks and traps in the fine print is over. This new bureau is based on the simple idea that if the playing field is level and families can see what's going on, they will have better tools to make better choices."
The new bureau was first proposed by Warren three years ago. In 2007, Warren, a professor at Harvard Law, proposed the creation of "a new regulatory body to protect consumers who use credit cards, home mortgages, car loans and a host of other products.
"Clearly, it is time for a new model of financial regulation, one focused primarily on consumer safety rather than corporate profitability," she wrote in an essay in Democracy: A Journal of Ideas.