The bill would delay pending IRS regulations aimed at limiting groups' political activities.
The Obama administration put the finishing touches on a new law that takes effect in July.
House Republicans next week will call up a bill that aims to reduce the proliferation of unfunded federal mandates on state and local governments, companies and other entities.
The House GOP will pass a legislative package aimed at federal regulations.
A string of school shootings is prompting renewed calls for more stringent federal rules.
The Kentucky Republican's legislation would apply to federal contests.
Sen. Dean Heller (R-Nev.) wants to change the operations at the Federal Communications Commission.
A new study found most countries do not have rules in place to address virtual currencies.
"The president doesn't have the power to waive the law," Rep. Tom Rice said.