WASHINGTON — President Barack Obama has repeatedly said he’s willing to work with Republican lawmakers on health care reform, and that they can meet with him any time they want to discuss new ideas.

“If you come to me with a set of serious proposals, I will be there to listen,” Obama said Thursday at a health care rally in Maryland, repeating a pledge he made directly to congressional Republicans a week earlier. “My door is always open.”

But is the door open to all Republicans?

Some say apparently not.

Republican U.S. Rep. Tom Price of Roswell, a doctor and chairman of the conservative House Republican Study Committee, said he has written the White House several times seeking an audience with Obama on health care, but his letters have gone unanswered.

“I can only conclude he’s being insincere,” Price said of the president.

Price sponsored a Republican health care proposal that calls for tax breaks to help the uninsured pay for coverage. The bill is one of at least 30 Republican proposals on health care, according to Price and others.

U.S. Rep. Phil Gingrey, a Republican from Marietta, also said he has requested time with the president to no avail. Gingrey has floated health care legislation pertaining to digital medical records and lawsuit liability limits for doctors.

“We can’t get close enough to the door to even see if it is open,” said Gingrey, who also is a doctor. “There are too many gates around it for us to even see it.”

That Obama hasn’t met with the two relatively low-ranking House Republicans from Georgia isn’t necessarily surprising.

But the president has rebuffed meeting requests from the highest-ranking Republicans, too.

At the office of House Minority Leader John Boehner of Ohio, spokesman Michael Steel said House Republicans haven’t met with Obama since April.

In May, Boehner and other House Republican leaders sent a letter to Obama asking for another meeting. Three weeks later, they got a response from the president thanking them but making no mention of a meeting, according to a copy of the president’s letter provided by Steel.

“We’ve been getting the door slammed in our faces,” Steel said.

White House officials say Obama has met “repeatedly” with members of both parties and that his health care plan draws from ideas from both Democrats and Republicans.

“The president remains deeply committed to working in a bipartisan fashion to pass health insurance reform,” spokeswoman Gannet Tseggai said. “President Obama ... has remained clear about his intention to continue working across the aisle.”

White House press secretary Robert Gibbs said at Thursday’s briefing that Obama met Wednesday with a group of senators that included Sen. Robert Bennett (R-Utah).

Obama has also met over the past several weeks with Republican Sens. Chuck Grassley of Iowa, Mike Enzie of Wyoming, Olympia Snowe and Susan Collins of Maine, and Bob Corker of Tennessee.

“I assume there will be other Republicans [who will meet with Obama] as this debate continues,” Gibbs said Thursday.

Part of the reason the president hasn’t paid more attention to Republican options may be because he doesn’t think they’ll solve the problems he’s trying to solve, said Liz Carpenter, an associate policy director at the New America Foundation, a left-leaning advocacy group.

Some GOP proposals might extend health care coverage, for instance, but they won’t reach Obama’s goal of covering every American, Carpenter said. Other Republican proposals might not do enough to rein in high insurance costs to satisfy his goals.

“I think the president wants to be bipartisan, but the president also wants to solve the problem,” Carpenter said. “And unfortunately, not every proposal solves the problem.”