"We shouldn't just allow this to happen and shrug it off. This is serious business. Those Latin American companies enjoy certain trade benefits with the United States. We ought to look at all of that to send a very clear message that we won't put up with this kind of behavior," the Michigan Republican told CNN's Candy Crowley on "State of the Union" on Sunday.
Rogers said the countries willing to accept Snowden are using the former intelligence worker as a "public relations tool."
Offers from Venezuela, Bolivia and Nicaragua
With the United States flexing its international clout, Snowden had faced a string of rejections to asylum requests. The Latin American countries were the first to respond in the positive.
Bolivian President Evo Morales, who became part of the saga last week when his presidential plane was denied permission to enter the airspace of several European countries amid a rumor about Snowden being aboard, said his country is "willing to give asylum."
Nicaraguan President Daniel Ortega has said he would grant Snowden asylum "if the circumstances permit."
"We are an open country, respectful of the right of asylum ... and it is clear that if the circumstances permit it, we would gladly receive Snowden and give him asylum here in Nicaragua," Ortega said last week.
The asylum offers could provide Snowden a chance to evade U.S. authorities, though it is unclear how he would get to Venezuela or the other countries.
In a speech Sunday, Cuban President Raul Castro said he supports the Latin American countries' rights to grant asylum.
"We back the sovereign right of Venezuela and all the states in the region to grant asylum to those persecuted for their ideals or fights for democratic rights, according to our tradition," he said.
Snowden's exit from Russia would provide relief to authorities there, who appear weary of the American's presence at the airport.
Alexei Pushkov, head of the lower Russian legislative body, the Duma, recommended Snowden leave the airport, where he has been holed up since June 23.
"Sanctuary for Snowden in Venezuela would be the best solution," Pushkov tweeted Saturday. "He can't live in at Sheremetyevo."
Bolivia's 'fair protest'
Bolivia's position on asylum follows outrage by its president over his sidetracked trip from Russia last week.
Several European countries refused to allow Morales' plane through their airspace Tuesday because of suspicions Snowden was aboard. With no clear path home available, the flight crew made an unscheduled landing in Vienna, Austria, where authorities confirmed Snowden was not a passenger.
Bolivia's asylum offer is a "fair protest" to the incident, which involved Portugal, Italy, France and Spain, Morales said. Spain has said it did not restrict its airspace.
On Monday, Bolivia summoned the four countries' ambassadors, demanding answers over what happened, Communications Minister Amanda Davila told reporters.
Irish court denies U.S. arrest warrant request
As the global guessing game over Snowden's next steps continues, the United States has asked a number of countries to arrest him and send him back to the United States if he sets foot in their territories.