According to Mr Barnes, Connally brought him to several Middle Eastern capitals to lobby regional leaders to convince Tehran that they would receive a better deal from Republican nominee Reagan if they kept hold of the hostages until after the election.
Mr Barnes told the newspaper that he was persuaded to come forward with his account by the news last month that Mr Carter is receiving home hospice care.
“History needs to know what happened,” Mr Barnes told the newspaper. “I think it’s so significant and I guess knowing that the end is near for President Carter put it on my mind more and more and more. I just feel like we’ve got to get it down some way.”
According to the newspaper, records at Lyndon Baines Johnson Library show that Connally and Mr Barnes left on a trip from Houston on July 18, 1980 to visit Jordan, Syria, Lebanon, Saudi Arabia, Egypt, and Israel, and returned to the US on Aug 11.
Mr Barnes said he was certain that the reason for the trip was to deliver the message to Iran to hold the hostages until after the election.
He claimed that when the pair met the first of a number of Middle Eastern leaders, Connally told them: “Look, Ronald Reagan’s going to be elected president and you need to get the word to Iran that they’re going to make a better deal with Reagan than they are Carter.”
Mr Barnes added: “He [Connally] said, ‘It would be very smart for you to pass the word to the Iranians to wait until after this general election is over.’ And boy, I tell you, I’m sitting there and I heard it and so now it dawns on me, I realise why we’re there.
“I’ll go to my grave believing that it was the purpose of the trip.”
Mr Barnes does not suggest that Reagan, who won two terms as US president, knew anything about the trip, but Mr Barnes told The New York Times that Connally briefed the chairman of Reagan’s campaign, William Casey, when they returned to the US.