A federal judge has put off a decision on whether to block President Trump’s firing of a Federal Reserve Board governor, Lisa Cook, until next week, when lawyers for the two sides have been told to present new arguments.

Mr. Trump announced this week on his Truth Social account his decision to oust the appointee of President Biden based on an allegation that she committed mortgage fraud.

Ms. Cook sued the president on Thursday, claiming he does not have the right to fire her. The Federal Reserve chairman, Jerome Powell, and the Board of Governors are also named in the suit.

Mr. Trump says he is firing Ms. Cook over so-far unproven allegations that she committed mortgage fraud when she purchased a home and condo in 2021, before she was appointed to the governing board by Mr. Biden.

By law, Fed governors can only be removed “for cause,” but it is not clear what constitutes sufficient cause.

In a United States District Court in Washington on Friday, Ms. Cook’s lawyer, Abbe Lowell, argued that the rationale for her firing was not legal and asked the judge to keep in her job while the matter is resolved.

There was a lengthy back-and-forth between the judge and an administration lawyer over the meaning of “for cause.”

Mr. Trump cited claims by the director of the Federal Housing Finance Agency, Bill Pulte, that Ms. Cook has committed mortgage fraud by simultaneously claiming two homes as her primary residence.

Mr. Lowell ridiculed the claims, telling Judge Jia Cobb, “You can’t have Director Pulte’s crazy midnight tweets be the cause.” 

At the end of Friday’s hearing, Judge Cobb gave both sides until Tuesday to submit more filings. It was unclear when the judge will rule on the request for an injunction.

The removal of Ms. Cook would allow Mr. Trump to replace her with someone more in line with his policies. The Federal Reserve, which sets interest rates for the country, has long been treated as an independent board tasked with keeping the economy growing without high inflation and encouraging a balanced labor market.

Mr. Trump has repeatedly attacked Mr. Powell over his handling of interest rates. The president had previously mused about firing the chairman but now seems content to wait to replace him when his term ends next year.

If Mr. Trump is able to replace Ms. Cook, his appointees will make up four of the board’s seven members, the Associated Press reported.