The U.S. Senate campaign of Linda McMahon is stepping up criticism of former U.S. Rep. Chris Shays, a tacit acknowledgement of the bounce that her rival for the GOP nomination is getting from a poll showing him in a tie with leading Democrats.

Shays is trying to turn the Republican primary into a test of electability in November, a message that some GOP leaders say is resonating with a party that holds no statewide or federal office in Connecticut.

“The winning candidate will have to demonstrate electability by any means possible,” said Jerry Labriola, the new Republican state chairman, who is publicly neutral in the race.

Linda McMahon

There is the sense that a door could be shutting on the GOP in Connecticut for a long time if McMahon or Shays cannot exploit the opportunity presented by the retirement of Sen. Joseph I. Lieberman after 24 years in Washington.

“This is a cycle where I am noticing more than ever a deep desire by our party’s rank and file to nominate the most fiscally conservative candidate and win — with an emphasis on winning,” Labriola said.

A Quinnipiac poll last week gave Shays new life, showing him in a virtual tie with either of the leading Democrats, U.S. Rep. Chris Murphy, D-5^th District, and former Secretary of the State Susan Bysiewicz.

Not only did it show Shays with surprising strength in general-election matchups, it fueled concerns about whether the high-water mark for McMahon was 43 percent, her share of the vote in the 2010 race against Democrat Richard Blumenthal.

McMahon’s best showing in general-election matchups came against the least-known contender for the Democratic nomination, state Rep. William Tong. The survey found her leading Tong, but with only 43 percent.

“I don’t care who you are, the Quinnipiac poll meant a lot,” said Lawrence F. Cafero Jr. of Norwalk, the Republican leader in the state House. “It meant a lot to the Shays campaign, and it meant a lot to the McMahon campaign.”

Cafero, who is uncommitted in the race, said the poll is shaping the strategy of the two campaigns.

McMahon, the World Wrestling Entertainment co-founder who spent $50 million of her own fortune losing two years ago, suddenly has shifted her attention from Murphy, the Democratic front-runner, to Shays.

On Tuesday, the McMahon campaign released an internal poll conducted Sunday and Monday that showed her with a 21-percentage point lead among Republicans, an attempt to restore a sense that she is the inevitable GOP nominee.

Chris Shays

Quinnipiac found Shays trailing McMahon by nine points among Republicans, compared to 15 points in the previous poll in September.

That was followed Wednesday by a counterattack on Shays over comments he made Sunday on WFSB’s “Face the State” suggesting that McMahon won the nomination in 2010 by paying some delegates as staff and vendors.

“Congressman Shays continues to disrespect the Republican Party, its leaders and Republican Town Committees across the state,” said Corry Bliss, McMahon’s campaign manager, in an email blast. “He is attacking the integrity of the nominating process and accusing our leaders of unethical behavior.”

In the email and also in an interview, Bliss accused Shays of hypocrisy, saying the former congressman has hired a state legislator and a Republican town chairman, while he is criticizing McMahon’s hiring.

“Forty-eight hours before the convention, our campaign will post a full list of delegates who serve as staff or vendors and their relationship to the campaign. We will also make this list available at the convention in Hartford,” Bliss said in his email. “And when Congressman Shays makes his list publicly available, I think people will be shocked to see that he is paying a minimum of one State Representative and a Republican Town Committee chair to work on his campaign. Congressman Shays is both hypocritical and disingenuous in his continued attacks on Linda.”

Both candidates say they are committed to transparency, even though the Republican State Central Committee recently rejected a proposal requiring candidates to disclose if delegates or other officials are on their payrolls.

“He exemplifies everything people don’t like about Washington,” Bliss said of Shays. “He thinks there is one set of rules for him and one set of rules for everyone else.”

Shays called the attack “silly.” Anyone on his staff will be wearing badges identifying them as campaign staff at the nominating convention in Hartford in May, and hiring a handful of elected officials who are not delegates is neither disingenuous nor hypocritical, he said.

He said his campaign has hired two state legislators, Reps. Brenda Kupchik of Fairfield and Penny Bacchiochi of Somers, and Robert Zappi, the town chairman of Westport. As of Dec. 31, Shays paid $10,000 to Kupchik and $6,369 to Bacchiochi, according to campaign finance reports.

Bacchiochi, who is a field director for Shays, said she always makes clear she is on staff when speaking on Shays’ behalf.

Labriola downplayed the exchange Wednesday, saying the McMahon and Shays campaigns both expressed support of the disclosure resolution and have agreed to abide by its terms, even though it was rejected by the state central committee.

“The fact they have now unilaterally signed onto these proposals is a positive sign,” Labriola said. “Transparency is paramount to ensuring a fair and open nominating process. I am confident that we will hold the most open and transparent state convention in history on May 18 in Hartford.”

McMahon’s opponent in the 2010 primary, former U.S. Rep. Rob Simmons, R-2^nd District, had accused Labriola’s predecessor, Chris Healy, of favoritism. Healy’s wife was on McMahon’s campaign staff.

“Going forward under my watch as state chairman,” Labriola said, “there will be no questions as to the integrity of our nominating process.”