Other possibilities include Ohio Senator Rob Portman, former Minnesota Governor Tim Pawlenty, rising Republican Congressman Paul Ryan and New Jersey's bluntly spoken Governor, Chris Christie.
But speculation is all it is. The Romney campaign prides itself on discipline and ability to keep a secret, and the candidate told reporters that, apart from him, the only person who knows who is on the list is his trusted aide Beth Myers, his chief of staff when he was Governor of Massachusetts.
But there have been some clues to Romney's thinking. The biggest of these is his repeated insistence that his running mate will be someone who is ready and qualified to take over as president if necessary.
Romney will bend over backwards not to repeat the mistake of Senator John McCain, the last Republican nominee, who selected the flashy but woefully unqualified Sarah Palin.
The choice cast serious doubt on McCain's judgment, and almost certainly contributed to the size of his defeat.
Under the constitution, a vice-president's formal duties are few - they come down to presiding over the Senate, where he casts the deciding vote in the event of a tie - and many have chafed at the limitations.
In practice, a vice-president's job and influence (or lack of them) are determined by the president himself. Lyndon Johnson, for instance, was virtually excluded from John Kennedy's inner circle. But under the inexperienced George W. Bush, Dick Cheney ran what amounted to a parallel administration, especially during Bush's first term.
A vice-president's biggest responsibility is the unspoken one: the possibility that he might have to step into the top job at a moment's notice.
Nine of the country's 44 presidents have been vice-presidents catapulted into the Oval Office by the death or resignation of their boss, most recently Gerald Ford after the fall of Richard Nixon in 1974.
By this yardstick, Rubio, the son of Cuban immigrants, may be found wanting. Despite his undoubted charisma, he has been a US senator for barely 18 months. Lack of experience could also rule out Christie and Bob McDonnell, Governor of Virginia, both of whom only took office in 2010.
With Jeb Bush, the former Florida Governor, having taken himself out of consideration, the most obvious front-runners are Portman, a senator and former congressman who served as George W. Bush's budget director, and Pawlenty, a two-term Governor of the swing state Minnesota, who has a popular touch that Romney lacks.
But nothing is certain, not even the date of an announcement. Most often, the choice is made on the eve of the convention, this year in late August. But Romney, who is cautious and well organised, may act sooner, perhaps as early as next month.
VICE SQUAD
Those who made a difference, for better or worse:
Lyndon Johnson: The Texan LBJ may have been the last vice-presidential nominee who helped in terms of votes, shoring up the wobbling Democratic south for John Kennedy in 1960.
Spiro Agnew: In 1973 he became only the second vice-president in history to resign, after admitting taking bribes when he was Governor of Maryland. Agnew's fate prefigured that of his boss Richard Nixon nine months later.
Sarah Palin: The notion of the pitifully unqualified Palin sitting the proverbial "heartbeat from the presidency" undoubtedly was a factor in John McCain's loss to Barack Obama.
Dick Cheney: Cheney was little electoral help to George W. Bush in 2000, but became America's most powerful and divisive vice-president.
Al Gore: Bill Clinton's deputy was the first "modern" - more influential - vice-president.
Dan Quayle: As vice-president to George H.W. Bush, Quayle became a figure of fun (famously even misspelling the word "potato" in a school class.)
- Independent
