"I can't see how you can negotiate a comprehensive immigration agreement over the weekend," said Gutierrez, who led the Commerce Department under former President George W. Bush. "It takes a plan. ... You're not going to do it over the weekend."
Trump warned that he would impose a 5% tariff on all Mexican goods starting Monday if Mexico doesn't take action to substantially curb the number of migrants crossing the U.S. border. The tariffs, if enacted, will go up incrementally to 25% by October.
Top Mexican officials met with White House officials on Wednesday in a bid to stop the tariffs. Trump tweeted Wednesday night from Europe that "Progress is being made, but not nearly enough!"
Talks are to resume Thursday.
If the tariffs are implemented, Gutierrez said, GOP officials are likely to push back after the initial 5% levy.
"I do think the president is going to get a lot of pushback," he said on "Squawk Box." "We have overplayed our hands on tariffs, and I think everybody has had enough."
Senate Republicans have signaled this past week they oppose the duties.
"There is not much support in my conference for tariffs, that's for sure," Senate Majority Leader Mitch McConnell told reporters Tuesday.
Former presidential candidate Sen. Ted Cruz, R-Texas, said he was worried about how the duties would impact Texas' economy. Last year, U.S. imports from Mexico surpassed $346.5 billion and exports topped $265 billion.
"If the outcome of this game of chicken is massive new tariffs that destroy jobs in Texas and across America, that would be a terrible outcome," Cruz said.