Sen. Rand Paul (R-Ky.) says AIPAC's supporters would back his legislation.
The letters were spearheaded by the American Israel Public Affairs Committee.
"There should not be a vote at this time on the measure," U.S.'s main pro-Israel lobby said.
The White House has said a "zero enrichment" option is unrealistic.
House and Senate leaders will host a joint meeting with the Israeli leader on Monday.
A pro-Israel group on Friday denied a report that it had agreed to hold off on publicly pressing...