Supporters of Mitt Romney are pushing back against calls for a campaign shake-up from Rupert Murdoch and Jack Welch, two industry titans who recently took to Twitter to advise the Republican presidential candidate to dump his staff.

Murdoch was among 50 people who met with the former Massachusetts governor at the Union League Club in New York City on Thursday afternoon.

Murdoch, the chief executive of News Corp., which publishes the Wall Street Journal and operates Fox News Channel, tweeted Sunday that Romney’s campaign team is insular and outmatched by President Obama’s.

“Met Romney last week. Tough O Chicago pros will be hard to beat unless he drops old friends from team and hires some real pros. Doubtful,” Murdoch wrote.

Welch, a former chief executive of General Electric, later tweeted a statement in support of Murdoch’s position.

But according to Kellyanne Conway, a Republican strategist who also attended the meeting, Murdoch’s issues with the campaign staff were never discussed.

“I can tell you confidently and competently that neither Governor Romney or anyone in that room raised any talk of personnel, let alone a shake-up,” Conway said Tuesday. “Personnel matters were not discussed whatsoever. It was a very constructive conversation where he showed some teeth and a little leg. To me, Romney seemed like that guy in the Florida debate all over again.”

New York Jets owner Woody Johnson and Randy Falco, the chief executive of Univision, were among the guests. The Supreme Court’s decisions on health care and immigration were among the topics, Conway said. The meeting was first reported by Politico.

Romney, who is vacationing in New Hampshire, is standing by his team, which includes a mix of longtime aides, such as Eric Fehrnstrom and Beth Myers, who is heading the search for Romney’s running mate.

“Governor Romney respects Rupert Murdoch and also respects his team and has confidence in them,” Romney press secretary Andrea Saul said.

Murdoch’s grumblings raised questions about the staff and whether prominent business leaders are souring on the campaign even as they pour millions of dollars into the candidate’s war chest.

At the meeting, Murdoch pressed Romney and his aides to get tougher on Obama and asked about Romney’s stance on immigration. He later tweeted his thoughts in response to a follower who said Romney has brains but needs more stomach and heart: “Romney has all these and more, but just to see more fight. And Hispanics a surrender to O. Cn not afford, hurts senate too.”

Polls show that Romney is lagging far behind Obama among Latino voters. At the meeting, the Republican insisted that he will maintain his position on immigration rather than risk appearing to switch sides. He also mentioned Sen. Marco Rubio (Fla.) as a top surrogate who will help him attract support among Latinos.

The calls by Murdoch and Welch for a staff shake-up came as Fehrnstrom, a top campaign aide, sided with Democrats who said that Obama’s health-care reform law penalizes rather than taxes people who don’t buy insurance. Romney has made the same argument about the health-care plan he signed into law when he was governor of Massachusetts.

Republicans, however, have used the tax argument — part of Supreme Court ruling that upheld the law — to try to reframe the health-care debate and tag Democrats as tax-and-spend liberals.

Fehrnstrom’s comments seemed to take that issue off the table.

Murdoch later tweeted: “Romney people upset at me! Of course I want him to win, save us from socialism, etc but should listen to good advice and get stuck in!”