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 Olin GrahamTop Louisiana health official rips Cassidy over ObamaCare repeal bill Senate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Overnight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions 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 PortmanRobert (Rob) Jones PortmanWeek ahead in tech: Debate over online sex trafficking bill heats up 'Hillbilly Elegy' author won't run for Senate Brown, Portman urge Trump administration to move quickly on a steel decision 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 Margaret CollinsSenate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Ryan: Graham-Cassidy 'best, last chance' to repeal ObamaCare Collins skeptical of new ObamaCare repeal effort MORE (R-Maine), a key centrist vote, along with budget hawks like Sens. Bob CorkerRobert (Bob) Phillips CorkerOvernight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions Senate passes 0B defense bill Corker pressed as reelection challenges mount MORE (R-Tenn.) and Tom CoburnTom Coburn-trillion debt puts US fiscal house on very shaky ground Al Franken: 'I make fun of the people who deserved it' The more complex the tax code, the more the wealthy benefit 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 Andrew BoehnerSpeculation mounts, but Ryan’s job seen as safe Boehner warns Trump: Don't pull out of Korea-US trade deal GOP Rep: Ryan wasting taxpayers dollars by blocking war authorization debate 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 McConnellAddison (Mitch) Mitchell McConnellSenate passes 0B defense bill Overnight Health Care: New GOP ObamaCare repeal bill gains momentum Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea 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 Dean BluntTop Senate Dem: We're going forward with understanding we can work with White House on DACA Sunday shows preview: Trump officials gear up for UN assembly Air Force One is Trump’s new boardroom 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."