President Obama is hoping to win over Republican lawmakers on a budget deal through personal appeals, placing a series of phone calls this week in hopes of rallying bipartisan support.

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On Tuesday, Obama phoned Sen. Lindsey GrahamLindsey GrahamTrump’s feud with the press in the spotlight Senators eye new sanctions against Iran Republicans play clean up on Trump's foreign policy MORE (R-S.C.), a key conservative voice in the upper chamber. Graham told NBC News he was "very encouraged" by the brief conversation, a sign the president's outreach efforts to top Republicans could be gaining traction.

Graham also told reporters that he would be open to increased revenues — a key aspect of the president's proposal — if it was accompanied by a significant overhaul of entitlement programs like Medicare and Social Security.

Earlier this week, Obama spoke with Sen. Rob PortmanRob PortmanRyan tries to save tax plan Rift in GOP threatens ObamaCare repeal Overnight Tech: GOP split on net neutrality strategy | Trump's phone worries Dems | Bill in the works on self-driving cars MORE (R-Ohio), the former White House budget director under President George W. Bush, who is considered among his party's top financial experts. Obama also reportedly spoke with Sen. Susan CollinsSusan CollinsPruitt sworn in as EPA chief Comey meets Intel senators amid uproar over Trump-Russia ties EPA breaks Twitter silence to congratulate new head MORE (R-Maine), a key centrist vote, along with budget hawks like Sens. Bob CorkerBob CorkerRepublicans play clean up on Trump's foreign policy GOP Congress unnerved by Trump bumps Trump makes nuclear mistake on arms control treaty with Russia MORE (R-Tenn.) and Tom CoburnTom CoburnCoburn: Trump's tweets aren't presidential The road ahead for America’s highways Rethinking taxation MORE (R-Okla.).

White House press secretary Jay Carney said Tuesday that Obama hoped to work with legislators "interested in common-sense solutions."

"The president is engaging with lawmakers in the hope that we can move forward in a balanced way, because he believes and knows that there are Republicans who agree with the vast majority of the American people -- with the majority of Republicans in the country, with the majority of independents in the country -- that we should do this in a balanced way, that we should include revenues, as well as entitlement cuts, spending cuts, when we further reduce our deficit, and so he's looking for solutions," Carney said.

Carney also hinted that the president's more aggressive tact may be borne out of frustration with Republican leadership in Congress.

"You know, thus far we have not seen from the leadership, an interest in taking up balance as an approach to dealing with our fiscal challenges," Carney said.

The feeling is likely mutual. Following the "fiscal-cliff" negotiations in January, Speaker John BoehnerJohn BoehnerFormer House leader Bob Michel, a person and politician for the ages Former House GOP leader Bob Michel dies at 93 Keystone pipeline builder signs lobbyist MORE (R-Ohio) pledged that he would no longer engage in one-on-one negotiations with the White House. Last month, Senate Minority Leader Mitch McConnellMitch McConnellRepublicans play clean up on Trump's foreign policy Americans brimming with optimism on the economy McCain hopes Americans can be confident GOP-controlled Congress can investigate president MORE (R-Ky.) told reporters her was "not interested" in last-minute negotiations to avert the sequester.

“Read my lips: I’m not interested in an 11th-hour negotiation," McConnell said.

Obama has not spoken to McConnell in his latest push, although he did host the Senate leader at the White House on Friday to discuss the sequester. But the Republicans who have been contacted have seemed generally positive when discussing the administration's effort.

"It's a start," Sen. Roy BluntRoy BluntJudiciary Committee wants briefing, documents on Flynn resignation Intel Dem: House GOP now open to investigating Flynn Dems: 'Crazy' to trust GOP to investigate Flynn MORE (R-Mo.) told The Wall Street Journal on Monday.

A spokesman for Collins echoed the optimism.

"Sen. Collins said they had a good discussion about the need for a bipartisan agreement on several critical issues including the unsustainable, $16.6 trillion debt and sequestration," Kelley said. "She encouraged further discussions of a substantive nature."