The collapse of a bipartisan Senate plan to end the funding dispute could leave 4,000 furloughed FAA employees and about 70,000 laid-off airport construction workers without paychecks until after Labor Day, the aides said.
The aviation authority would also lose an estimated $1.2 billion in ticket-tax revenue, officials said.
"This just does not make sense," Sen. Kay Bailey Hutchinson, R-Texas, told The Washington Post, breaking ranks with House and Senate Republicans.
"Is that fiscal responsibility? I don't think so," she said. "It doesn't pass the smell test."
The deadlock is over a ruling by the National Mediation Board -- an independent government agency that coordinates labor-management relations within the railroad and airline industries -- that favored union efforts to organize airlines.
Republicans want to reverse an NMB rule that union-organizing elections should be decided by a simple majority of those who vote, the Post said. Instead, they want to go back to a decades-old rule that says in airline union-organizing elections, the votes of eligible voters that are not cast should be counted as voting "no."
Sen. Orrin Hatch, R-Utah, denounced the agency for creating the simple-majority rule, calling the agency a "runaway National Mediation Board."
Hatch blocked a Democratic effort to extend FAA funding until September.
The Democratic-controlled Senate and the Republican-controlled House are also at odds over proposals to cut rural air-service subsidies.
House members began heading home Monday after passing the federal budget-debt bill and Senators were expected to follow suit Tuesday or Wednesday after they vote.
"Congress should not leave on its summer vacation until they finish an FAA [funding] bill," Transportation Secretary Ray LaHood said Monday. "Don't hold hostage innocent people who are not getting their pay check."