If approved by the Senate, Ms Sotomayor would join Justice Ruth Bader Ginsburg as the second woman on the current Supreme Court.
The BBC's Richard Lister, in Washington, says it appears likely she will be confirmed by the Senate - but only after some tough questioning during the confirmation process.
Conservative activists have already challenged comments she made a few years ago that a judge should not dismiss their own gender or ethnicity in deciding cases, our correspondent says.
Mitch McConnell, leader of the Republicans in the chamber, said his colleagues would treat Ms Sotomayor fairly, but stressed that the Senate should not be a "rubber stamp".
"We will thoroughly examine her record to ensure she understands that the role of a jurist in our democracy is to apply the law even-handedly, despite their own feelings or personal or political preferences," he said.
White House officials told the Associated Press that Ms Sotomayor would bring more judicial experience to the Supreme Court than any other justice confirmed in the past 70 years.
Mr Obama had stressed the need to appoint a justice who had life experience as well as legal capabilities.
Ms Sotomayor was raised on a housing estate in the Bronx - in one of New York City's most deprived areas.
She is said to have been inspired by the Perry Mason TV series to become a judge, and was educated at both Princeton and Yale.
She has been nominated to serve as a judge by both Republican and Democrat presidents.
In 1991, George H W Bush named her a district judge and his successor in the White House, Bill Clinton, nominated her to the circuit court in 1997.
The Supreme Court is the highest court in the US, and has the authority to strike down any state or federal law it deems unconstitutional.
Each of the nine Supreme Court justices is nominated to the post for life.
Of the incumbents, four are regarded as liberal, four as conservative and one is seen as unaligned.
If appointed, Ms Sotomayor is expected to be on the liberal wing - leaving the political balance unchanged as Mr Souter is also a liberal.
US MEDIA REACTION TO THE SOTOMAYOR NOMINATION
Latino groups have been grumbling somewhat about their representation (or lack thereof) in the Obama administration, as well as the fact that immigration reform doesn't appear to be on the White House's front-burner. But this pick buys Obama A LOT of time with Hispanics - a demographic he won last year, 67%-31% - on immigration and other issues. Is it a coincidence that Obama this week heads out West to Nevada and California, two states with large Latino populations?
For someone who has all the usual qualifications to be a Supreme Court Justice, she also has an unusual life story... It's the kind of story that makes you feel good about America and that still resonates as quintessentially American even though social mobility in the United States isn't quite what we like to think.
Since she is certain to be confirmed, there are plenty of smart conservatives who will, by midday Tuesday, have done the political cost-benefit analysis: at a time when Republicans are trying to demonstrate that their party can reach beyond rich white men, what mileage is there in doing anything but celebrating such a historic choice?
I am... not favorably impressed with her notorious statement that "a wise Latina woman with the richness of her experiences would more often than not reach a better conclusion than a white male who hasn't lived that life." Not only is it objectionable in and of itself, it also suggests that Sotomayor is a committed believer in the identity politics school of left-wing thought. Worse, it implies that she believes that it is legitimate for judges to base decisions in part based on their ethnic or racial origins.
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