Former Hewlett Packard CEO and 2016 presidential hopeful Carly Fiorina is reportedly eyeing up a new political run – this time to replace Reince Priebus as chair of the Republican National Committee – although some fear her refusal to back Donald Trump may hurt her chances.

Fiorina, who was largely seen as an unlikely candidate for the GOP nomination when she announced her candidacy last year, surprised many commentators with a strong run.

In particular, a moment at the September debate in California, in which Fiorina implored President Obama and Hillary Clinton to watch leaked videos that described controversial practices at Planned Parenthood, resonated with Republican voters and saw her briefly surge near the top of the pack in some polls. Her numbers later fizzled and she dropped out after a disappointing showing in the New Hampshire primary in February.

Fiorina has been spending the time since then campaigning for Republican House and Senate candidates, and has not endorsed Trump.

A number of outlets, including Time and Politico, reported Tuesday that the Fiorina is mulling a bid to replace Priebus after the November election – and is reaching out to state party chairs offering to help “in any way” as a way of laying the groundwork for a run.

Time reported that domain names CarlyForChair.com and CarlyForRNC.com were bought up at the same time in July.

Priebus has not yet announced if he will seek another term as party chair.

Should Trump win the White House in November, he would have considerable say over who becomes chair, and so would be unlikely to recommend Fiorina. But should Hillary Clinton win the White House, her chances look much stronger.

While Fiorina has never held elected public office, she does have some experience in party leadership. After her unsuccessful 2010 California Senate bid, she served as vice-chair of the National Republican Senatorial Committee and later the chair of the American Conservative Union Foundation. Her background in management and technology industries could also help her clinch the role.

Some Fiorina supporters say it would be the right move for her, and for the party.

“She has proven herself to be an outstanding, eloquent spokeswoman for conservative principles and the Republican Party, as someone who had been on the front line implementing those principles in the real world and in the private sector,” Keith Appell, former senior adviser to the “CARLY for America” Super PAC, told FoxNews.com.

“One of the things she did well as a candidate was getting a lot of people to nod their heads and say ‘yes’ when she explained the practical applications of conservative policies and principles. That’s something the party has needed,” Appell said.

However, some believe her refusal to back Donald Trump could be fatal, even if the billionaire loses. Fiorina briefly became Ted Cruz’s running mate before he dropped out, and said of Trump in March: “he does not represent me, he does not represent our party.”

Former Iowa Republican Party chair Craig Robinson told FoxNews.com that while Fiorina is a good candidate for the job, her apparent gamble that Trump will lose shows she is not a team player.

“Where I have questions is on the timing of all this. It seems she is plotting her next personal step at a time when Republicans at large should be doing what they can to help their nominee win the White House,” Robinson, who also runs The Iowa Republican blog, said. “She’s out there running for RNC chair instead of helping the team elect Donald Trump president.”

Robinson said that although “I can’t think of anyone who does a better job of communicating the message against Hillary Clinton,” he said that putting herself above the party may cost her the job.

“I think she really misplayed this. I think she’s going to turn off certain elements within the RNC committee that she’s going to need for support,” he said.  “They want to win and they’re supporting Donald Trump. If she’s not doing that I don’t see how these individuals say she’s right to lead the party.”

Fox News' David Lee Miller contributed to this report.

Adam Shaw is a Politics Reporter and occasional Opinion writer for FoxNews.com. He can be reached here or on Twitter: @AdamShawNY.