Obama will need lawmakers to approve an ambassador and provide funds for an embassy.
The White House candidate said Obama has not made human rights a priority.
The House appears unlikely to finish work on appropriations before the August recess.
Senators, including Marco Rubio, Ted Cruz and Rand Paul, are asking for a plan by July 15.
Under fire from conservatives, the legislation was pulled from the floor in February.
Harriet Tubman was previously picked in an online poll to replace Andrew Jackson on the $20 bill.
The Republican presidential hopeful said Jerusalem should have a U.S. embassy before Havana.
The Republican leader added that a constitutional amendment likely wouldn't pass Congress.
The Florida Republican is the latest lawmaker to voice their concern over the talks.
Rep. Phil Roe wants to ensure that workers at the Department of Veterans' Affairs who violate civil law aren't eligible for bonuses.