"The letter has been a surprising controversy that came up," said Sen. Mark Kirk (Ill.), a moderate Republican up for reelection in 2016, who signed the letter.
Sen. Jerry Moran (R-Kan.), who also signed on, called the negative response "a significant overreaction" to "a number of senators expressing their view of something that's a fact, that Congress, the Senate, needs to be involved."
Speaking at a breakfast with Bloomberg reporters and editors, Sen. Ron Johnson (R-Wis.) said, "I suppose the only regret is who it's addressed to,'' adding, "But the content of the letter, the fact that it was an open letter, none whatsoever."
Forty-six fellow Republicans signed Cotton's letter, which was designed to derail a potential nuclear deal with Iran. The letter warned Iranian leaders that any lasting deal must get the input of Congress, not just President Obama.
No Democrats signed the letter. Obama has criticized it, and so have many other Democrats.
"I'm opposed to Iran having a nuclear weapon. So I'll do everything I can to keep that from happening, and that was one of the steps," said Sen. Dean Heller (R-Nev.), another signatory.
Asked whether he was surprised at the backlash, Heller responded: "The liberal backlash? No."
But it's not just the left that has denounced the strategy. The Wall Street Journal editorial board took issue with it. And Rep. Peter King (R-N.Y.) reportedly expressed reservations.
"Everybody's entitled to their opinion," said Sen. John McCain (R-Ariz.), who signed the letter.
"I think an assertion of Congress's constitutional prerogative on something like this was probably a good thing," said Sen. John Thune (R-S.D.), who signed the letter.
Seven Republican senators declined to sign the letter. One of them, Sen. Susan Collins (Maine), said that she did not expect a major uproar but that that was not why she kept her name out.
"I just felt that it was not an appropriate letter and it was not the job of the Senate to be communicating with the Iranians during such a sensitive time," she said.
Asked whether she sensed any regret from any of her colleagues who had signed the letter, Collins said: "You'd have to ask them."