The pall that hung over the Capitol last week after Congress failed to do anything about the sequester has lifted — if only a bit.

President Barack Obama’s outreach to Republicans this week has revived flagging hopes for a grand budget bargain even though there has been no breakthrough on the key question of taxes.

Obama has called more than a half-dozen Republicans, dined with 12 and had lunch Thursday with House Budget Chairman Paul D. Ryan, R-Wis., and ranking member Chris Van Hollen, D-Md. And there’s more to come, including a visit to each GOP caucus in the Capitol next week.

Plus, the Republicans who Obama hopes will help form a “caucus of common sense” are ignoring blowback among some conservative activists and groups.

Sen. Lindsey Graham, R-S.C., has led the charge to thaw the relations between the White House and the GOP — compiling the list of a dozen Republicans who dined with the president Wednesday night at the Jefferson Hotel at the president’s request.

Graham again held out hope Thursday for a grand bargain including a tax overhaul deal that would lower rates and help pay down the debt. And he dismissed criticism he’s gotten for reaching out to the president.

“Let me put it this way … if I can’t go have dinner with the president of the United States to talk about the problems that face our nation, I shouldn’t be running,” said Graham, who is up for re-election next year. Graham added, “The enemies of this nation … are al-Qaida, the Taliban and the terrorists. I have political opponents on the other side of the aisle. They are not my enemies.”

The meal included a few minutes of chitchat but quickly grew serious as the lawmakers hashed out their thoughts on Social Security, Medicare and taxes, with the president and the lawmakers alike noting the urgency of getting something done in the next several months.

Sen. Ron Johnson, R-Wis., a tea party favorite, said the parties remain at an impasse despite the dinner, but he still has hope that a deal may be reached that would focus largely on the need to change entitlements. And he said he’s not worried about criticism from the right.

“Let’s face it, neither side’s going away. So, if we’re going to fix these entitlement programs it’s going to have to be a bipartisan approach, and you can’t come up with a bipartisan approach if you are not sitting down and talking to people,” Johnson said.

Johnson said he could support dramatic tax change, perhaps with creative triggers attached that could help bridge the divide between the parties and generate new revenue.

“You could have triggers … I’d certainly be open to that,” he said. He also said that if a far simpler tax code can be written that reduces the massive costs of complying with the existing code, he’d support the government getting some of those savings.

But Johnson said Republicans feel they already gave ground in the fiscal-cliff debate. “I think he kind of recognizes that,” he said of the president.

The party’s base, meanwhile, is suspicious of the optimism following the dinner, with many on Twitter questioning senators for dining with Obama, and conservative groups warning lawmakers not to go wobbly on taxes.

“The only reason for optimism is if the President promised that tax increases are completely off the table,” Michael A. Needham, CEO of Heritage Action, said in a statement. “A fine dining experience with the President of the United States may be enjoyable, but until he jettisons his flawed, economically damaging approach to deficit reduction, there is little reason for excitement.”

Sen. Lamar Alexander, R-Tenn., who has been urging the president for years to start talking to lawmakers, also expressed new hope.

“We’ve grown exhausted talking to each other. … We need Moses to lead us, that’s what presidents are supposed to do,” he said, suggesting that tackling the debt could be Obama’s “Nixon-to-China” moment.

And Alexander dismissed the idea that grass-roots conservatives would punish senators for dining with the president. “Conservatives in Tennessee want to fix the debt,” he said. “You can’t fix the debt unless the president leads the way. And for him to lead the way, there are going to have to be these conversations.”

The president’s outreach, meanwhile, made some on the left nervous that once again he might, in their eyes, give up too much to the GOP.

Sen. Bernard Sanders, I-Vt., has already ripped the president for being willing to cut Social Security and veterans benefits by adopting a stingier gauge of inflation.

“The president historically has not been a strong negotiator,” he said. “The fear is he is going to get a deal that will probably not be strong enough on revenue — that does worry some of us.”

And liberal Sen. Tom Harkin, D-Iowa, said he is not worried about where the president’s heart is, but he is “concerned about his advisers — and their commitment to our Democratic priorities and whether they are willing to — in order to reach some kind of agreement — to pitch some of those over the side. … I cast no aspersions on anyone, but when you go from Wall Street to here to Wall Street to here you get a distorted view of the world.”