President Obama has invited a small group of fiscally-conscious House Democrats -- known as the "Blue Dogs" -- to the White House Tuesday morning in an effort to win their support for his controversial health care reform package, FOX News has learned.

"We're just not there yet," said Rep. Baron Hill, D-Ind., one of the lawmakers invited to the session with Mr. Obama. "We're getting there."

When asked what the main concern was, Hill responded, "Time."

Hill is a member of the House Energy and Commerce Committee, a key panel struggling to approve a health reform bill. Many conservative Democrats on the committee are already shell-shocked from criticism they've heard for voting for a climate and energy bill in June. Many of those lawmakers are reluctant to take a tough vote again.

Hill's rural, southwest Indiana district is a case study in the red-blue divide.

His winning percentages reflect the tenuous grasp he has on his seat and the vulnerabilities he faces on contentious issues. Voters first elected Hill to the House in 1998 with 51 percent of the vote. He then narrowly defeated Republican Mike Sodrel with 51 percent in 2002. But in 2004, Sodrel rallied to unseat Hill by a mere 1,500 votes. Hill returned the favor two years later, capturing 50 percent of the tally against Sodrel in a three-way contest. He again beat Sodrel in 2008.

To secure his vote on health care, Hill said he's seeking specific cost-cutting and wants a better understanding about how "bundling" health care dollars would work. Under the current fee-for-service system, health insurance companies divide up payments among doctors, technicians and nurses. Bundling would tie all payments together.

"He hasn't taken anything off the table," Hill said of the president. The Indiana Democrat signaled that there are about 12 policy areas that need alteration to satisfy the Blue Dogs.

The Blue Dogs are moderate to conservative Democrats who represent historically Republican turf. They advocate low taxes and restraint in government spending, and the Blue Dog Coalition is particularly spooked about the trillion dollar price tag of the president's health care proposal and a ballooning of the federal deficit by $236 billion. They dislike tax increases which would soak the rich and feel that Democratic leaders are racing to okay a bill by the August recess.

"It's clear in the fact that they don't have the votes with the language they've got now," said fellow Blue Dog Rep. Allen Boyd, D-Fla.

Boyd said he'd like to see the House take a break on health care and punt to the Senate. The reason is that political chasms over health care are much wider in the Senate than they are in the House. The thinking is that any Senate-passed bill would stand a better chance of scoring approval in the House.

"Stop. Let them work. See what they can do," Boyd said of the Senate.

But despite Boyd's skepticism, he's not ready to quit on health reform.

"Any thing can happen," he said."This place is known for miracles."