The provision would require certification that Iran supports Israel's right to exist.
Senators are discussing which amendments to consider on the bill.
Republicans said they are "perplexed" by the administration’s threat assessment.
The Democratic leader used Barrasso as an example of Republicans being "relentless" on Obamacare.
McConnell pledges to take up Obama's trade push after the Iran bill.
The majority leader said it was "appropriate" given Tuesday's events that senators were debating Iran legislation.
It was the first in a series of politically tricky amendments on the Iran deal.
The freshman Arkansas Republican is a first-time dad.
"I would just urge all people to exercise restraint," Ben Cardin said.