As the end of current Fed Chair Janet Yellen's term approaches in February, eyes on Wall Street are seeking signs of who President Trump might appoint to succeed her in the post. There are not yet clear signals of who Trump will tap to head the central bank, or if he might even re-nominate Yellen, but that hasn't stopped observers from weighing in.

Treasury Secretary Steven Mnuchin told Bloomberg this week the Trump administration hasn't yet made any decisions on the next Fed chair, "whether we're going to have a new one or not going to have a new one."

Trump has to decide about whether to keep Yellen on when her term expires in February or seek someone new. The president criticized Yellen on the campaign trail but since taking office has softened his tone, telling the Wall Street Journal in April he likes the "low-interest rate policy" she has maintained.

"I think the likelihood is low, but not zero [that Trump will renominate Yellen], and higher than the likelihood that a traditional Republican economist outside the president's orbit would get the nod," said Wharton financial historian Peter Conti-Brown.

He said figures such as John Taylor and Glenn Hubbard are likely out of the running, given they're much more hawkish than any sitting president would prefer unless he had a profound commitment to fighting inflation. 

Taylor is a Stanford economist for whom the Taylor rule for interest rates is named, and Hubbard is the dean of Columbia University's Business School who served as Deputy Treasury Secretary under President George Bush and chairman of the Council of Economic Advisors under George W. Bush.

"Nothing about President Trump's public statements or persona suggest he is that person," he said. "More likely, I think, is one of Trump's closer associates with little expertise in monetary economics. I would include Gary Cohn in that category. An excellent and successful banker, yes, but no obvious expertise in central banking."

The former Goldman Sachs (GS) - Get Report executive is helming the search for the next Fed chief, The Wall Street Journal reported last week, increasing speculation he might seek the post himself.

"Gary Cohn...does understand commodities and currencies and global financial assets, and he brings that sophistication," said Judy Shelton, a conservative economist and adviser to the Trump campaign.

Cohn wouldn't be the first non-economist to lead the Fed. Truman appointee Thomas McCabe was an industrialist whose only previous central banking experience was as a director of the Philadelphia Fed. Carter pick G. William Miller came from a corporate background as well. "He was way in over his head and was essentially fired by promotion, removed from his position to serve as Secretary of the Treasury," Conti-Brown said.

But appointing someone so close within his orbit could open Trump up to scrutiny.

"There are always issues if the Fed chairman is seen to be someone who works for the president, I don't know if that's generally a good thing, so that might be a reservation," Shelton said.

Conti-Brown pointed to Arthur Burns, Fed Chair under Richard Nixon and Gerald Ford, as a historical example of a too-close-for-comfort central banker. "Burns was a leading economist, but had by that time converted into a Republican Party man under Eisenhower and Nixon," he said. "More than anything, he defined the role as subservient to the president. If Trump appoints a close associate like Cohn, watch Republicans and Democrats alike raise this historical parallel."

Republicans and Democrats alike resisted George W. Bush's attempt to appoint Harriet Miers to the Supreme Court due to her inexperience as a judge and closeness to the president. Her nomination was eventually withdrawn.

"If I were advising the president, I would strongly recommend Yellen herself. She has shown that she's not dogmatic, which is what any president will want: someone willing to be accommodative if there's evidence for it, rather than reflexively hawkish," Conti-Brown said. "I also think there is very good reason to be continue the long tradition of cross-partisan appointment that has been in place for almost 40 years. But I don't think Trump will take that advice, nor do I think those close to him are giving it."

Shelton said she would like to see "someone who is not afraid to challenge the conventional wisdom" at the Fed, pointing specifically to John Allison, former president of the libertarian Cato Institute and former CEO of BB&T Corporation (BBT) - Get Report . "He understands the impact of bank regulation on small business lending," she said. "He's a serious contender, a serious guy."

Shelton, the director of the Sound Money Project at Atlas Network and a proponent of returning to the gold standard, has been floated by some as a potential contender to replace Yellen as well. She acknowledged she wouldn't mind the job but thinks it's unlikely.

"I'd love that, but I'm sort of tagged with this gold bug idea," she said.

Editors' pick: Originally published June 23.