Appearing on WFSB-TV's "Face the State," McMahon said that because she won more than 60 percent of the 1,208 delegates on Friday, Shays should quit so Republicans can be unified against Democrats in the fall.
"That's a pretty strong statement to Republicans in the state, that I am the endorsed candidate, and I would really hope that Congressman Shays would really look at and consider that statement and understand how more effective it would be if we were a unified party moving forward now to take on Congressman Murphy."
U.S. Rep. Chris Murphy won the Democratic Party endorsement May 12 and will face former Secretary of the State Susan Bysiewicz in a primary Aug. 14
Shays, a former U.S. representative, said he will remain a candidate. In a separate appearance Sunday on the TV program, he said many delegates supported him because he is committed to challenging McMahon in the primary.
"The first stage was for me to get in the race. The second stage was for me to qualify for a primary and now I'm in the primary," he said. "There are two people in the primary."
"I had some real courageous delegates who were willing to support me because I agreed to be in the primary," Shays said.
The two candidates also disagreed on what role former Republican Gov. John G. Rowland would play in the campaign. Rowland resigned in a corruption scandal in 2004 and was sent to prison for 10 months.
McMahon ruled out a campaign role for him.
"I might appear on his radio show," she said, referring to his program on WTIC-AM.
Shays said he would welcome Rowland's backing. He credited Rowland and former U.S. Rep. Nancy Johnson, a fellow Republican, for helping him win election to the U.S. House in 1987.
"I'd love to have his support. He's paid his price," Shays said.
Mike Clark, a former FBI agent and one-time Republican candidate in Connecticut's 5th congressional district race, filed a federal elections complaint over payments to Rowland for consulting work for a company owned by the husband of another candidate, Lisa Wilson-Foley.
She said Rowland was a volunteer adviser to her campaign and the payments were unrelated to her House bid.
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