WASHINGTON (Reuters) - As Republican presidential candidate John McCain starts thinking through who he wants as a vice presidential running mate, he may find that many potential picks carry risks as well as rewards.

Republican presidential candidate Sen. John McCain (R-AZ) (R) gestures for the cashier to keep the change after paying for a slice of pizza at Verrazano Pizza during a campaign stop in the Brooklyn borough in New York April 10, 2008. REUTERS/Shannon Stapleton

Political experts say McCain has many difficult choices to make on a running mate. If he pick a conservative, he risks alienating the moderate voters he prizes. If he picks a centrist, he should brace himself for the conservative fallout.

Take for example ex-Massachusetts Gov. Mitt Romney, whom McCain defeated for the Republican presidential nomination. Romney has made clear he would not mind being No. 2.

But social conservatives remember that Romney supported abortion rights in Massachusetts before switching to a more conservative stance in his presidential race.

“If Gov. Romney is on your ticket, many social conservative voters will consider their values repudiated by the Republican Party and will either stay away from the polls this November or only vote down the ticket,” said a letter sent to McCain and signed by 20 leading social conservatives.

Secretary of State Condoleezza Rice has been in news reports so much recently as a vice presidential possibility that a WNBC/Marist poll found McCain and Rice would defeat, in New York state, a Democratic ticket of Barack Obama and Hillary Clinton.

Rice denies she is interested in what would be her first political race. McCain told ABC’s “The View” on Thursday that he had not talked to her about it.

“I have not, but all of us admire Condoleezza Rice. She’s a great American. She’s been a role model to all of us,” he said.

EASY TARGET

Rice would carry risks, however, as an easy target for Democrats who accuse McCain of basically mimicking the policies of unpopular President George W. Bush.

Republican strategist John Feehery said that Rice, who is black, could help balance a McCain ticket against either Obama, who would be America’s first black president, or Clinton, who would be the first woman president.

“It would be a historic choice,” he said. “But I just don’t sense that it’s based on reality.”

What about some of the other Republicans whom McCain defeated for the nomination? They also carry baggage.

Former New York Mayor Rudy Giuliani was never trusted by conservatives for his positions on abortion and gay rights. Former Arkansas Gov. Mike Huckabee is considered too inexperienced to be a heartbeat away from the presidency.

Former Tennessee Sen. Fred Thompson never really energized crowds on the campaign trail.

“There are a deep bench of presidential candidates but a thinner bench for vice presidential candidates,” said Ken Weinstein, a political expert at the Hudson Institute. “A number of the presidential candidates would be tough sells.”

McCain gets along famously with Connecticut Independent Sen. Joe Lieberman, a moderate who would appeal to independents. But conservatives remember Lieberman as the vice presidential running mate for Democrat Al Gore in 2000.

GOVERNORS BECKON

A number of Republican state governors are believed to be on McCain’s list of 20-or-so candidates, such as Minnesota Gov. Tim Pawlenty, Florida Gov. Charlie Crist and South Carolina Gov. Mark Sanford. There is also former Ohio Rep. Rob Portman, who was Bush’s budget director.

None of them has a particularly high national profile. But they could help McCain capture a battleground state in the November election.

Grover Norquist, head of the conservative group Americans For Tax Reform, said McCain should reduce the importance of the vice presidential pick by first naming people he would appoint to other critical positions.

For example, he could say a conservative, former Texas Republican Sen. Phil Gramm, would be his treasury secretary, and conservative lawyer Ted Olson would lead McCain’s effort to find federal judges.

In the end, however, there is a simple requirement for McCain, according to Todd Harris, a Republican strategist who was a McCain spokesman in his 2000 presidential bid.

“Regardless of which way he goes, I think he should pick someone who he is personally comfortable with, because unlike so many in Washington, he is not particularly good at faking friendships,” Harris said.