Grassley tied his move to delays over his questions about Clinton's emails and a top aide.
The GOP leader added the report shows the IRS's "seething contempt" for conservative groups.
The Foreign Relations Committee chairman suggested lawmakers could take it up later this year.
Senators are expected to return to Washington on Sept. 8.
John Cornyn suggested the push on the floor was part of a political manuever by Dems.
The Maine senator said he had "never faced a more difficult decision."
The Senate will turn to the nuclear deal when it returns from the August recess.
Cory Bennett 08/05/15 03:15 PM EDT
The move means the cyber bill is highly unlikely to get a final vote before recess.
The American Democracy Legal Fund accuses Ron Johnson of violating the Senate's Internet policy.
Sen. Chris Murphy said it's "pure fantasy" to think Congress can force a better deal.