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 GrahamDHS chief takes heat over Trump furor Overnight Defense: GOP chair blames Dems for defense budget holdup | FDA, Pentagon to speed approval of battlefield drugs | Mattis calls North Korea situation 'sobering' Bipartisan group to introduce DACA bill in House 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 PortmanFlake's anti-Trump speech will make a lot of noise, but not much sense Top GOP candidate drops out of Ohio Senate race Overnight Tech: Regulators to look at trading in bitcoin futures | Computer chip flaws present new security problem | Zuckerberg vows to improve Facebook in 2018 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 CollinsDemocrats search for 51st net neutrality vote Overnight Tech: States sue FCC over net neutrality repeal | Senate Dems reach 50 votes on measure to override repeal | Dems press Apple on phone slowdowns, kids' health | New Android malware found Overnight Regulation: Dems claim 50 votes in Senate to block net neutrality repeal | Consumer bureau takes first step to revising payday lending rule | Trump wants to loosen rules on bank loans | Pentagon, FDA to speed up military drug approvals MORE (R-Maine), a key centrist vote, along with budget hawks like Sens. Bob CorkerRobert (Bob) Phillips CorkerSenate campaign fundraising reports roll in Congress should take the lead on reworking a successful Iran deal North Korea tensions ease ahead of Winter Olympics MORE (R-Tenn.) and Tom CoburnThomas (Tom) Allen CoburnRepublicans in Congress shouldn't try to bring back earmarks Republicans should know reviving earmarks is a political nightmare Former GOP senator: Trump has a personality disorder 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 BoehnerDems face hard choice for State of the Union response Even some conservatives seem open to return to earmarks Overnight Finance: Trump, lawmakers take key step to immigration deal | Trump urges Congress to bring back earmarks | Tax law poised to create windfall for states | Trump to attend Davos | Dimon walks back bitcoin criticism 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 McConnellSessions: 'We should be like Canada' in how we take in immigrants NSA spying program overcomes key Senate hurdle Overnight Finance: Lawmakers see shutdown odds rising | Trump calls for looser rules for bank loans | Consumer bureau moves to revise payday lending rule | Trump warns China on trade deficit 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 BluntGOP senators eager for Romney to join them Senate GOP wary of ending Russia probes, despite pressure GOP on precipice of major end-of-year tax victory 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."