“Senators who would like to see this bill strengthened, as I would,” Mr. McConnell said last week, “will have that chance during a robust amendment process that we’ll soon have.”

That promise could conflict with Mr. McConnell’s presumed desire to see a bipartisan bill that forces the White House to accept some congressional role. Many Democrats who support the underlying bill said emphatically last week that they would abandon a bill with any legislative ornaments they disliked.

“I think the bill that came out of the Senate Foreign Relations Committee was a terrific and important and hard-won compromise,” said Senator Chris Coons, Democrat of Delaware, who enthusiastically supported the bill in the committee. If largely partisan amendments become attached to it, he said, “I would no longer support it.”

Senator Tim Kaine, Democrat of Virginia, who has been a strong advocate for Congress having a role in evaluating sanctions and other national security issues, said on the Senate floor Thursday, “I hope, as we get into deliberations on the floor next week, that this would be the spirit of all the colleagues who tackled this most important matter.”

Of course, not every amendment filed will make it to the floor. Senators sometimes file amendments to make a point without actually offering them. Mr. McConnell may get a deal to limit the number of amendments. While some, like the measure that would require the recognition of Israel by Iran, will no doubt entice many members, others have no shot of attracting the 60 votes they most likely need.

Procedure and math make many Democrats nervous. Amendments, such as one concerning Israel, that get the 60 required votes with the help of Democrats, and that might lead those same Democrats to pass the underlying bill, could prompt a veto from President Obama. But, in that case, the 67 votes may not be there for a veto override, which some Democrats suspect Mr. McConnell of relishing.