They unanimously backed sanctions if Obama can't verify that a deal is being followed.
The Senate is voting on amendments to the GOP budget.
Sen. Blunt said his amendment would "send a clear message" to the administration.
The bill would apply to detainees released after Feb. 1 of this year.
Sen. Scott said his amendment would raise awareness about a tax on premiums.
Only a small group in the House opposed the $200 billion legislation.
Senators backed year-round Pell grants for college students, but rejected a funding increase.
Senators rejected the measure in a 45-55 vote.
The paid sick days amendment was offered by Sen. Patty Murray (D-Wash.).
"America does not project power from bankruptcy court," he said.