WASHINGTON -- The aftermath of President Obama's televised question-and-answer session with House Republicans last Friday suggested it wasn't much of a success for the GOP. And the party has no intention of taking him up on a rematch, signaled Sen. John Cornyn (R-TX).

"We’re always happy to hear from the president but I don’t really feel any compelling need to do it [on camera]," Cornyn told Politico on Friday, when asked about the president's invitation to face off with Senate Republicans.

Pressed on how he reached his conclusion, Cornyn said: "For what purpose? Was it for photo op or is it serious? The president can invite Mitch McConnell, John Boehner or anybody he wants for a serious talk about issues."

Cornyn's stance contradicts the apparent willingness of some Republican lawmakers to face the president in a similar televised setting again.

"I would hope that he would repeat such a thing on a periodic basis," said Rep. Jeb Hensarling (R-TX). "I think it'd be great," added Rep. Paul Ryan (R-WI).

"I’ve always thought 'wow this is really a good tradition,'" remarked John Campbell. "[If] our presidents had to do that, that would really be a good thing."

The White House has repeatedly expressed interest in making it a regular feature.

"I can see why he and his handlers would want to replicate that," Cornyn said. "I think we’re more interested in serious public policy issues rather than providing another photo-opportunity for the president."

As the chair of the National Republican Senatorial Committee, Cornyn is a leading election strategist, so his opposition sheds light on how the GOP views the implications of a rematch on future electoral battles.

The first sign of Republicans acknowledging defeat last week came just after the event, when MSNBC's Luke Russert reported that GOP aides privately said "it was a mistake that we allowed the cameras to roll like that" and "we should not have done that."

In another sign of GOP defeat, Fox News cut coverage of the event halfway through -- considerably sooner than any other cable network, some of which broadcast the session in its entirety.

Democrats, by contrast, were very pleased with the results. The Atlantic's Marc Ambinder quoted "a very influential Democrat" as saying over e-mail, "I don't know whether to laugh or cry that it took a f$$@&$* year for Obama to step into the ring and start throwing some verbal blows."

"Accepting the invitation to speak at the House GOP retreat may turn out to be the smartest decision the White House has made in months," Ambinder remarked of the session.