Trump "has not made any decision" on when one will occur, Bill Pulte told Maria Bartiromo on  Fox Business Network Thursday morning, when she asked whether one could happen near term given Trump's recent post depicting an offering for the Great American Mortgage Corporation in November.

Trump has long been watching for the right timing for a Fannie-Freddie IPO since his first term, according to Pulte, who has rebranded FHFA as U.S. Federal Housing.

"He got approached to sell these companies for about $100 billion, if you can believe that, $150 billion today. That value is anywhere from $500 billion to $700 billion. I think it's going to $1 trillion, and potentially even higher," said Pulte, director of the Federal Housing Finance Agency.

The $500 billion-plus range is in line with a recent Bloomberg report citing Trump administration thoughts on valuation sought. Based on the information at hand at the time of Trump's post, Keefe, Bruyette & Woods analysts had pegged the value as closer to half that amount.

"Everybody wanted to steal this company from the American taxpayers and President Trump deserves great credit for saying, 'No thank you, we are going to do what's in the best interest of people,'" Pulte added in his Fox interview Thursday.

Pulte pointed to Fannie and Freddie's combined $7.8 trillion in assets and what he views as strong international demand for their shares, saying both factors could drive an "oversubscribed" offering.

Interest rates and related policy could be a factor

Comments later in the interview suggest Federal Reserve Chairman Powell's reluctance to raise interest rates, which both Trump and Pulte have been critical of, could be a consideration in getting maximum value for Fannie and Freddie in the market.

Low rates can be a mixed blessing for the mortgage industry and the GSEs. They can, for example, increase the enterprises' loan acquisitions and guarantee fee income but they also may reduce future opportunities to refinance and create new loans.

Pulte said the fed funds rate should be "zero or 1% at the highest" as opposed to above 4%. Trump administration officials have signaled little interest in pursuing Fannie and Freddie reforms that could push mortgage rates higher.

Powell, an independent policymaker speaking at the Jackson Hole Economic Symposium this week, has been cautious about lowering the fed funds rate, citing concern it could exacerbate inflation in prices for many items consumers buy, including shelter.

He also has said his role with respect to housing is limited given most mortgages are pegged to how longer-term bonds trade than the short-term rates the Fed controls.

Pulte has taken issue with that stance given that Fed policy and its outlook still can exert considerable influence on mortgage rates. He told Bartiromo on Thursday there could be "damage to the housing market up to two years after he's gone" in reference to Powell.

How a public offering would be structured still undetermined

As far as what form a public offering could take, Pulte said that GSE reform is a priority for Trump, the housing regulator also said there hasn't been a call yet on what or if anything will be done.

"President Trump has exercised great judgment and he'll exercise, in my view, that same judgment in deciding whether or not to do something with Fannie and Freddie," Pulte said.