A new Associated Press-GfK poll suggested that Americans have a more positive view of her than they did for any of former President George W. Bush's nominees to the high court. Half backed her confirmation.
As Sotomayor made her Senate debut with a series of private meetings, Republicans said they would prefer holding hearings on her nomination in September, which could cloud the speedy summertime confirmation Obama wants.
Sotomayor, who would be the high court's first Hispanic and its third woman, told senators she would follow the law as a judge without letting her life experiences inappropriately influence her decisions.
"Ultimately and completely, a judge has to follow the law no matter what their upbringing has been," Sen. Patrick Leahy, D-Vt., the Judiciary Committee chairman, quoted the nominee as saying in their closed-door session.
Republicans are questioning how she would apply the law, noting her remark in 2001 that she hoped her decisions as a "wise Latina" would be better than those of a white male who hadn't had the same experiences. Obama has said she misspoke; some Republicans have called the comment racist.
Leahy, hoping to shepherd a smooth and quick confirmation for Sotomayor, asked her what she meant by her 2001 comment and said the judge told him: "Of course one's life experience shapes who you are, but ... as a judge, you follow the law."
Sen. Jeff Sessions of Alabama, the top Republican on the committee, said Sotomayor used similar words with him as well, but he appeared to come away from the meetings unconvinced about her approach and whether she would be an "activist" who tried to set policy from the bench.
"We talked about the idea and the concept of personal feelings and ... how that influences a decision, and how it should not," Sessions said, declining to elaborate on the private discussion. Sessions, who is to meet Wednesday with Leahy to discuss scheduling Sotomayor's confirmation proceedings, said he thought hearings should wait until September - more than a month after Obama and Senate Democrats had hoped to have Sotomayor confirmed.
The exchanges came as Sotomayor rushed from one hotly anticipated meeting to another on Capitol Hill - 10 in all - visiting senators who will decide her future. She meets 10 more Wednesday.
In public, the nominee and senators were all grins and polite exchanges. Sotomayor chatted with Leahy about his grandchildren and smiled demurely as Senate Majority Leader Harry Reid praised her. Sessions called their talk "a delight."
But behind closed doors, they touched on the weightiest matters of the law and judging.
Democrats praise Sotomayor's life story as the New York-born daughter of Puerto Rican parents who was reared in a housing project and went on to Princeton and Yale before ascending to the highest legal echelons.
"We have the whole package here," said Reid, D-Nev. "America identifies with the underdog, and you've been an underdog many times in your life, but always the top dog."
A pack of journalists camped outside Reid's office hoping for a glimpse of the Supreme Court nominee this morning, but Sotomayor entered his office up a back elevator where video cameras are not allowed, reports CBS News Capitol Hill producer Jill Jackson.
A quick photo-op was permitted at the very top of the meeting, Jackson notes, but Sotomayor did not say a word. "That's normal for a Supreme Court nominee," she notes. "You typically don't hear them speak publicly until the actual Supreme Court confirmation hearing. Senators usually do the talking." (Read more from Jackson in the Hotsheet blog.)
In the new poll, half said she should be seated on the court while 22 percent opposed her confirmation. About a third had a favorable view of Sotomayor while 18 percent viewed her unfavorably.
Questioned about affirmative action, 63 percent support it for women and fewer, 56 percent, favor affirmative action for racial or ethnic minorities. The poll did not define affirmative action.
She was looked upon more positively than any of three Supreme Court nominees President George W. Bush put forward over four months in 2005: Chief Justice John Roberts, Justice Samuel Alito and Harriet Miers, who withdrew from consideration.
Roberts, the most popular of the three in polling at the time, was supported for confirmation by 47 percent, and 25 percent had a favorable impression of him.
Sotomayor's 2001 speech has inspired sharp rhetoric from some Republicans. Radio host Rush Limbaugh and former House Speaker New Gingrich have both branded Sotomayor a racist, and Limbaugh said choosing her for the high court would be like nominating former Ku Klux Klan leader David Duke.
Leahy called the criticism "among the most vicious attacks that have been received by anybody" and said given the tone, it would be irresponsible to wait until September for hearings that will give her a chance to respond.
Democrats hope to begin the sessions next month, which would meet Obama's goal of having her confirmed before the Senate departs in early August for a monthlong vacation.
But Reid said while Democrats want to hold hearings "as quickly as we can," they would not seek to impose "arbitrary deadlines."
Democrats hold 59 votes in the Senate - more than enough to win Sotomayor's confirmation - but short of the 60 it would take to advance the nomination should Republicans try to block it. Leading Republicans including Sessions have said they don't see doing so, but they are facing calls from conservative leaders to try to prolong the process by engaging in a long debate on the Senate floor.
While GOP senators have steered clear of tough language, they have in their own way questioned whether Sotomayor would bring a personal agenda to the bench.
"We need to hold our fire until we examine all of these opinions and writings," said Sen. Jon Kyl of Arizona, the No. 2 Republican. "The one clear thing that is becoming an issue ... is the question of the basis for making decisions."
Leahy told reporters he asked the judge whether he could repeat publicly what she told him privately during their meeting about how her personal experiences would shape her rulings.
Leahy quoted Sotomayor as saying, "There's not one law for one race or another. There's not one law for one color or another. There's not one law for rich, a different one for poor. There's only one law."
The White House, choreographing Sotomayor's interactions with the Senate, called the first meetings productive.
"We believe that she will get a fair shake and a fair set of hearings," presidential spokesman Robert Gibbs said.
At the same time, he kept up pressure on Republicans to allow a confirmation vote before the Senate's August recess.
The White House says it wants Sotomayor seated by September so that she can take part in discussions about which cases the high court will hear when the new term begins in October. There is a political component, too. Should her confirmation drag on through the traditionally slow news month of August, it could give opponents more time to try to derail it.