WASHINGTON – Georgia U.S. Sen. David Perdue hasn't shied away from bashing his Senate colleagues and party leaders for big-ticket policy foibles in recent months, but the freshman Republican is staying tight-lipped about his own future in the upper chamber.

Perdue said the Senate’s fumble on Obamacare repeal legislation shows the GOP has a “leadership issue” and a “membership issue,” in a Thursday interview on C-SPAN’s “Newsmakers” that will air on Sunday. But he dodged a question about whether he’d ever try to join the Republican leadership team in order to make changes from the inside.

“That’s flattering,” he said. “I’ve never heard – I mean I hear people talk about it – but right now we have a leader. We have leadership.”

Perdue's name has come up as a potential successor to Senate Majority Leader Mitch McConnell among some disaffected corners of the conservative universe agitating for the Kentucky Republican to step down. Jenny Beth Martin of the Tea Party Patriots recently indicated she wouldn't be opposed to Perdue as Senate leader.

And Perdue has shown his own frustrations with his party leader, directly leveling a shot at McConnell in September after the GOP failed to coalesce behind a last-ditch Obamacare repeal bill.

But Perdue's rhetoric was much more pared down on Thursday.

He credited McConnell for twice listening to his calls to keep the Senate working longer hours to confirm President Donald Trump’s nominees and make progress on other issues. He was sharper toward Susan Collins, John McCain and Lisa Murkowski, the three Republican senators who opposed recent Obamacare repeal plans, for not being team players on health care.

But when it came to his own political ambitions, Perdue said he saw aiding the implementation of Trump's agenda as his "only priority" at the moment.

"This is a situation where we have a leadership right now, and (in) my role as the only Fortune 500 CEO in Congress I try to add value on the topics that I have experience in the real world like this tax debate," he said. "I’ve lived with that, and so we’re able to bring some value to that conversation, I think, and that’s my role right now in the Senate."

Perdue has long been viewed as politically ambitious, but his name usually popped up as a potential candidate for Georgia governor or national office, not for a higher-profile Senate role. But Trump's ascendancy to the White House has shifted the discussion among some of his allies.

Perdue's remarks came as outside figures such as former White House adviser and Breitbart News executive chairman Steve Bannon escalated calls for McConnell to step down. Among Senate Republicans, however, McConnell's standing is considered strong.