Barack Obama, having claimed a majority of pledged delegates in the Democratic nominating contest, is looking to run out the clock against Hillary Clinton in the final two weeks of an exhaustive primary season.

He told supporters at a rally in Tampa, Fla., Wednesday, that the nomination is almost his, and his top strategist said the campaign is moving toward "closing out this process officially."

"We are at the threshold of being able to attain this nomination," Obama said in Florida.

Though he lost Kentucky to Clinton by a gaping 37 points in Tuesday's primary, the delegates he picked up with his second-place finish there and his first-place finish in Oregon put him within 70 delegates of the 2,026 it takes to win.

The message out of the campaign is that the race will be over soon, and it's time to gear the party toward facing presumptive GOP nominee John McCain.

But Clinton was resolute Wednesday in arguing she is still in the race and that delegates from the disputed Florida and Michigan primaries should be counted. She won those contests, but the delegates were stripped because the states held early primaries in violation of party rules.

"We believe the popular vote is the truest expression of your will," she said Wednesday in Boca Raton, Fla., attempting to draw clear comparisons between the current penalties on Florida and the state's disputed 2000 presidential vote that gave George W. Bush the White House. "The lesson of 2000 here in Florida is crystal clear. If any votes aren't counted, the will of the people isn't realized and our democracy is diminished.

"You didn't break a single rule and you should not be punished for matters beyond your control," she said.

However, while Clinton campaigned in Florida as a primary candidate, Obama campaigned as a general election contender, devoting much of his time to striking at McCain.

Obama strategist David Axelrod said it's "very likely" the Illinois senator will reach the nomination threshold by June 3, when Montana and South Dakota hold the final primaries. Puerto Rico votes June 1, and with 55 delegates holds the most weight of the remaining three primary contests, giving the island rare influence this election year.

Obama's campaign proudly claimed Tuesday night that it had achieved a majority of pledged delegates, hoping that "milestone" will send a signal to fence-sitting superdelegates that it's time to pick a side -- preferably theirs.

Spokesman Robert Gibbs called the achievement "an important marker for superdelegates" who have said they would recognize the the person with the most pledged delegates as the winner.

So far, one superdelegate has moved into Obama's column since Tuesday night -- Connecticut Rep. Joe Courtney, whose district voted for Clinton in the state's Feb. 5 primary, which was won by Obama. Clinton too gained a superdelegate on Wednesday -- Craig Bashein, a member of the Ohio delegation to the national convention in Denver this August.

Superdelegates are party insiders and officials who are not tied to the outcome of state contests, while pledged delegates are assigned proportionally based on election results.

As of Wednesday afternoon, Obama had a total of 1,962 delegates to Clinton's 1,779 delegates, according to Associated Press tallies.

With victory near, Obama is very publicly reaching out to Clinton, nudging her and her supporters toward unifying the party against McCain.

"Senator Clinton has run an outstanding campaign and she deserves our admiration and respect because she has set the standard," Obama said in Tampa. "She has broken barriers and will open up opportunities for a lot of people, including my two young daughters.

In Des Moines, Iowa, Tuesday night, he said: "Some may see the millions upon millions of votes cast for each of us as evidence that our party is divided, but I see it as proof that we have never been more energized and united in our desire to take this country in a new direction."

The Kentucky primary, while not halting his march toward the nomination, underscored the difficulty Obama has attracting key voting blocs -- particularly white, working-class voters -- and the resistance some Democrats have in accepting Obama as the nominee.

Even as Gallup tracking polls show Obama pulling away from Clinton in national surveys, FOX News exit polls in Kentucky showed just 50 percent of Democrats there would support Obama in the fall if he is the nominee.

Clinton has offered no sign to supporters that it's time to abandon their fight for the nomination.

Though the New York senator would have to win almost all the remaining pledged delegates and superdelegates to wrest the nomination from Obama, she is still holding out hope for some windfall of delegates out of Florida and Michigan.

Clinton is eagerly awaiting the May 31 meeting of the Democratic National Committee Rules and Bylaws Committee, which will determine whether to lift the penalty, at least partially, on Florida and Michigan Democrats.

Clinton is also arguing to superdelegates that she is the better candidate to take on McCain in the fall. A Rasmussen poll on Wednesday found that in head-to-head match-ups in Florida, Clinton beats McCain 47-41, while a match-up between McCain and Obama would result in a Republican victory of 50-40.

Just 57 percent of Florida Democrats said they would vote for Obama in November, and 27 percent said they would vote for McCain. The two candidates are essentially tied among unaffiliated voters.

In an e-mail Wednesday to supporters, Clinton blasted media pundits for declaring a premature finish to the race.

"Yesterday, voters in Kentucky joined you in sending a clear message -- this campaign is far from over," the e-mail said.

Speaking of her Michigan and Florida delegate campaign, she said Tuesday night: "I am going to keep making our case until we have a nominee, whoever she may be."

Click here to read Major Garrett's analysis of Obama's ascent to the top of the Democratic field.

FOX News' Bonney Kapp and The Associated Press contributed to this report.