Unveiling a new state task force to combat infectious diseases on Monday morning, Perry called for federal officials to implement "enhanced screening procedures" at "all points of entry" to the United States and create "fully staffed quarantine stations" wherever people are entering the country.
"Washington needs to take immediate steps to minimize the dangers of Ebola and other infectious diseases," said the governor.
A pair of other Republicans called Monday for Obama to appoint a single adviser to coordinate the government's response to Ebola. Others have gone even further, calling for flight bans from West African countries and raising concerns about catastrophic scenarios. Republican strategists say it is all part of an effort to flex leadership credentials and tap into concerns Americans have with President Obama's readiness to handle crises after a series of missteps in his second term.
"Republicans are emphasizing the mishandling of the Ebola crisis by the Obama administration and tying it to the theme of government incompetence," said Republican strategist Ron Bonjean. "After a string of failures by the White House, the latest crisis over Ebola containment further underscores the need for new leadership."
Greg Mueller, a veteran of three GOP presidential campaigns, said he thinks recent lapses in security and safety have spurred Republicans to speak out about Ebola.
"We've got people walking though the front door of the White House, we've got child trafficking on the border, we've got [the Islamic State] beheading people and now you've got Ebola," he said.
After meeting with top advisers about Ebola at the White House Monday afternoon, President Obama said his administration will be "working on protocols to do additional passenger screening both at the source and here in the United States." The White House said it was not considering a travel ban.
Thomas Eric Duncan, who is fighting to survive in a Dallas hospital after being diagnosed with Ebola last week, flew from Liberia to Dallas last month. There are very few direct flights from West Africa to the united States, so Duncan changed planes in Brussels and at Dulles before arriving in Dallas -- news that has prompted concerns about air travel facilitating the spread of Ebola.
Sen. Jerry Moran (R-Kan.) and Rep. Frank Wolf (R-Va.) wrote Obama a letter Monday asking him to appoint a "single, senior advisor who will be responsible for coordinating all U.S. agencies and policies involving the international and domestic response to Ebola. They suggested Obama bring in former George W. Bush administration officials Colin Powell, Robert Gates and Mike Leavitt to "assist with this serious challenge."
Some possible Republican presidential hopefuls have focused their calls on tightening travel restrictions.
Louisiana Gov. Bobby Jindal (R) has called for a halt to flights from Ebola-stricken countries. Sen. Ted Cruz (R-Tex.) has penned a letter to the head of the Federal Aviation Administration with questions about what steps the agency was taking to prevent the spread of Ebola through air travel ahead of the busy holiday season. Sen. Rand Paul (R-Ky.) raised concerns in a recent radio interview about the the possibility "a whole ship" of American soldiers could get Ebola.
The Republican calls for new measures to prevent the spread of Ebola came as a Pew Research Center poll released Monday showed Republicans are more skeptical than Democrats about the federal government's ability to prevent an outbreak of Ebola.
"There's only so much that a state can do," said Perry.
The issue has also cropped up in several key midterm contests in states with some of the busiest international airports in the United States.
Thom Tillis, the Republican nominee for Senate in the crucial battleground state of North Carolina, last week called for U.S. officials to ban travel from Liberia, Sierra Leone and Guinea. In Michigan, Republican Senate nominee Terri Lynn Land has also called for a travel ban.
North Carolina is home to Charlotte Douglas International Airport, which saw more than 20 million passengers in 2012, according to the Department of Transportation. In Michigan, Detroit Metropolitan Airport saw nearly 16 million passengers that year.
At least one Democrat has mentioned Ebola in a campaign ad. Sen. Mark Pryor (D-Ark.) released a commercial in August attacking Rep. Tom Cotton (R) for cutting "billions from our nation's medical disaster and emergency programs."
Mark Berman contributed to this post.
Read more: How the Ebola epidemic sped out of control