Sen. John Thune (R-S.D.) said "a death in the family should not be a taxable event."
Senators have been voting since noon Thursday as part of the budget amendment process.
Sen. Ed Markey said the amendment undercuts President Obama's authority.
The amendment would have blocked $1.2 trillion in Medicaid cuts.
Sen. Chris Coons said Congress must pay for the fight against ISIS.
The Budget Committee leader said the chamber has taken 17 votes in more than six hours.
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.