Story highlights

Congressional Republicans are upset with the Obama administration's new policies with Cuba

They hope to blunt his influence by either defunding a potential embassy or blocking an ambassador nominee

It's unclear how effective they can be even when they take control of Congress in January

(CNN) —  

President Barack Obama’s move to relax travel, trade and economic restrictions with Cuba set off a firestorm from opponents on Capitol Hill, who immediately vowed to block the new policy.

Top Republican leaders and other key lawmakers were furious they were kept in the dark about the secret mission orchestrated by the White House to bring home Alan Gross, an American hostage held in Cuba for the last five years. Although they supported Gross’ release they denounced the Obama administration’s sweeping changes to a policy that isolated the communist country for over 50 years, arguing these steps would only make it harder to force a more democratic government in Cuba.

READ: Bells ring in Havana, anger erupts in Miami

What can Congress do to roll back Obama’s actions?

Defund a new U.S. Embassy in Cuba

Florida Republican Sen. Marco Rubio, whose parents emigrated from Cuba, said he would use every tool at his disposal – including using Congress’ power of the purse to block funding for a new U.S. embassy in Havana.

But Sen. Patrick Leahy, D-Vermont, who was part of the U.S. delegation that flew down to pick up Gross, batted aside the funding threat, noting there are already diplomatic personnel working in a U.S. “Interests Section” in Havana.

“The horse is out of barn,” Leahy said.

The spending bill that Obama signed on Tuesday night funds the State Department’s operations through September. That measure includes funding for the U.S diplomatic facility in Cuba’s capital.

GOP leaders are discussing whether to strip that money out as part of their debate on spending bills early next year.

Republicans will control both chambers of Congress next year and both top leaders – House Speaker John Boehner and Senate GOP Leader Mitch McConnell – made statements critical of the President’s actions.

In an interview on Wednesday the incoming Senate Majority Leader Mitch McConnell said he deferred to Rubio, effectively agreeing to fight back aggressively to reverse the administration’s Cuba policy.

“Sounds like the correct response to me,” McConnell told The Associated Press about Rubio’s pushback to the President’s new policy. “I think he knows more about this than almost anybody in the Senate if not everybody in the Senate and I wouldn’t differ with his characterization.”

Texas Sen. Ted Cruz said he wasn’t sure the votes were there in the Senate, but he would be part of an effort to try to unravel the new rules.

“I am hopeful that we will prevent a bad deal and a bad decision,” Cruz said on CNN.

Block the nomination of the U.S. Ambassador to Cuba

Obama instructed Secretary of State John Kerry to “to re-establish diplomatic relations that have been severed since January of 1961.”

But Rubio, who is expected to chair a key Senate panel that would consider a presidential pick for U.S. Ambassador to Cuba, immediately vowed to block any nomination.

Democrats again point to existing diplomatic personnel in place to keep an embassy running in Havana.

RELATED: The wait for Cuban beaches and cigars continues

“There are lots of countries where we don’t necessarily have a confirmed ambassador in place right now, but we have full diplomatic relations with them,” Rep. Chris Van Hollen, D-Maryland, said.

Retain current U.S. embargo

The President outlined a series of changes such as new travel rules, allowing American tourists to use debit cards while visiting Cuba, permitting relatives to send money to family members living in Cuba. But he admitted the current embargo passed by Congress restricts the scope of what he could do. He called on Congress to lift the embargo – something Rubio and other top Republicans said wouldn’t happen in the next Congress.

While he was confident Congress wouldn’t repeal the embargo, Rubio acknowledged there are limitations to what Congress can do to stop the President’s plans from moving forward.

“Many of the changes that have been today such as diplomatic relations fall within the purview of the presidency. My criticisms are largely based on the fact that these are unwise decisions,” Rubio said.

CNN’s Ted Barrett contributed to this story