By Steve Holland and Melissa Fares
NEW YORK (Reuters) - President-elect Donald Trump stepped up his search on Monday for a new U.S. secretary of state, with the focus on David Petraeus, a former U.S. military commander in Iraq whose mishandling of classified information led to his resignation as CIA chief in 2012.
"Just met with General Petraeus--was very impressed!" Trump said on Twitter shortly after Petraeus, a retired general, left an hour-long meeting with the Republican winner of the Nov. 8 election at Trump Tower in Manhattan.
Trump's consideration of Petraeus, who has also been mentioned as a contender for the top job at the Pentagon - adds a new layer of drama to his unusually public deliberations over the top diplomatic job for his administration.
Petraeus, who led international forces in both Iraq and Afghanistan, was sentenced to two years' probation and fined $100,000 last year for the unauthorized removal and retention of classified information.
He admitted sharing classified information with his biographer, with whom he was having an affair. The scandal forced Petraeus to resign from the CIA in 2012.
Petraeus said after meeting Trump that the New York businessman "basically walked us around the world" in their discussion. "He showed a great grasp of the variety of challenges that are out there and some of the opportunities as well," Petraeus told reporters.
Trump, who takes office on Jan. 20, plans to hold talks with Mitt Romney on Tuesday in his second recent meeting with the 2012 Republican presidential nominee.
A Republican source close to the transition team said it had appeared late last week that Trump was leaning toward choosing Romney as his secretary of state but that the appearance of Petraeus at Trump Tower suggested the president-elect was still undecided and casting a wider net for the position.
U.S. Senator Bob Corker of Tennessee, the chairman of the Senate Foreign Relations Committee, and former New York Mayor Rudy Giuliani are also in the mix for the job of America's top diplomat, Trump aides say.
Frances Townsend, a national security aide during the administration of Republican President George W. Bush, also met Trump on Monday.
Petraeus' past mishandling of classified documents is unlikely to be an obstacle to Trump offering him a top government post, said a source who has advised the transition team on national security. That is despite Trump harshly criticizing Democratic rival Hillary Clinton during the election campaign for using a private email server while she was secretary of state.
Trump often compared the prosecution of Petraeus with the lack of legal action against Clinton, who was investigated by the Federal Bureau of Investigation but never charged with any offense.
"Just based on his public statements, I think (Trump) sees Petraeus as a good man who made a mistake, who did a fraction of what other people have done and received a lot more punishment,"
Among the Trump transition team, Petraeus and Romney are supported by Republicans looking for a candidate with gravitas who can unify the party, the source said.
Giuliani is favored by loyalists who worked on the election campaign. Kellyanne Conway, his campaign manager, has said Romney would cause a backlash among his supporters.
Ideological conservatives hope Trump picks John Bolton, the former ambassador to the United Nations, the source said. Corker is well regarded, but some in the Trump camp do not want too many senators coming into the Cabinet, the source said.
Trump arrived back in New York on Sunday after several days in Florida for Thanksgiving. He caused a stir on Sunday when he alleged on Twitter that “serious voter fraud” had occurred in the presidential election in New Hampshire, Virginia, and California - states won by Clinton - but provided no evidence to back his assertion.
The White House said on Monday there had been no evidence of widespread election fraud in the presidential contest, and all three states rejected Trump's allegation.
New Hampshire's deputy secretary of state, David Scanlan, said: “There’s no evidence that we’ve seen that supports claims like that. Voter fraud does occur, but it occurs in isolated instances.”
California Secretary of State Alex Padilla called Trump's allegations unsubstantiated and Virginia Commissioner of Elections Edgardo Cortes said they were unfounded.
(Additional reporting by David Brunnstrom in Washington and Julia Harte in Washington; Writing by Alistair Bell and Roberta Rampton; Editing by Jonathan Oatis and Peter Cooney)