White House officials promise that President Barack Obama will be more engaged than ever with lawmakers and the public on health care this month and next as the legislation enters a critical phase, with both the House and Senate trying to move bills onto the floor before recessing for the summer.

“His schedule is going to be filled with engagements on health reform,— one White House aide said.

But Obama aides said they are perplexed by criticism from Capitol Hill and elsewhere that the president isn’t doing enough lobbying on the health care legislation, pointing to a string of private and public sessions the president has had with lawmakers to solicit support for his top priority.

In recent days, Senators and aides have suggested the president wade more deeply into negotiations in order to accelerate progress toward getting a bill to the floor. But White House officials insisted Obama has been continuously involved with the Senate Finance and Health, Education, Labor and Pensions committees as well as the House panels that released their combined version of health care reform legislation Tuesday.

“The president has been engaged with leaders on Capitol Hill who are molding health care reform to the extent he thinks he needs to be engaged,— one Obama adviser asserted.

White House officials pointed to meetings in recent weeks with members of the House Ways and Means Committee, Senate Finance Chairman Max Baucus (D-Mont.) and Sen. Chris Dodd (D-Conn.), who is leading the HELP Committee effort on the bill, a White House session with Sen. Ron Wyden (D-Ore.), who has his own bill, and many others. Baucus huddled with Vice President Joseph Biden at the White House last week while Obama was in Europe.

“The president has been on the phone and had people up to the White House all the time,— the White House aide said. “It’s just that not all of it has been made public.—

Several sources noted that Nancy-Ann DeParle, Obama’s designated point person on health care, pretty much goes to work each day on Capitol Hill instead of the White House.

But some Obama allies on Capitol Hill also said the White House could spur faster movement on the legislation, particularly in the Senate, if he came down more forcefully on specific issues that are slowing agreements.

House Democrats, who have produced legislation paid for in part by a tax increase on the wealthy, want “cover— in the form of an Obama endorsement for the idea as they try to sell the new tax to voters.

But the Obama White House is stocked with former aides to President Bill Clinton — and their strategy is deeply informed by what is seen as the heavy-handed approach during the failed Clinton health care effort. These officials said it will ultimately redound against the president if he starts giving orders.

“Lines in the sand are not useful when you are trying to get people to talk to each other,— White House health care spokesperson Linda Douglass said.

Obama aides noted that the president has given unmistakable signals about what he wants and doesn’t want in the bill. Obama has been clear, they said, in his advocacy for a public health insurance option and his antipathy toward taxing health benefits. Nevertheless, one Senate Democratic source suggested there was resentment that it ended up falling to Senate Majority Leader Harry Reid (D-Nev.) to instruct Baucus to stop toying with the benefit tax idea in order to curry GOP support.

And White House officials noted that Obama is “open— to the type of tax on the wealthy included in the House bill as a pay-for, though some Democrats said a stronger embrace would give them greater political immunity.

Obama held two sessions Monday evening that heralded the start of his intensive July and August health care lobbying.

He met with Speaker Nancy Pelosi (D-Calif.), House Majority Leader Steny Hoyer (D-Md.), Reid and Baucus. He joined others in the meeting in pressuring Baucus to get his mark done as soon as possible, according to sources familiar with the session.

After the leadership meeting, Obama gathered with a handful of Blue Dog Democrats in a session that focused on how to pay for the legislation. Among those at the meeting were Reps. Stephanie Herseth Sandlin (S.D.), Baron Hill (Ind.), Mike Ross (Ark.) and Jim Matheson (Utah), aides said.

The president is also stepping up what the White House said has already been a busy schedule on the health care soapbox, putting in a plug for the bill Monday during an announcement of his choice for surgeon general. He has granted the TV networks interviews today in order to tout his health care plans.

Emily Pierce contributed to this report.