Former President Donald J. Trump turned down Fox News’s invitation to debate Vice President Kamala Harris on the air this month.

But the Republican nominee will appear on the network next week for an unusual televised town hall, fielding questions from an all-female audience.

The event, announced by Fox News on Friday, will focus on “issues impacting women ahead of the election,” the network said, including abortion, day care, child care, health care and the economy.

It will not be shown live. The event will be taped on Tuesday evening, in Cumming, Ga., and air on Wednesday at 11 a.m. The moderator is the Fox host Harris Faulkner.

Mr. Trump’s Democratic opponent, Ms. Harris, is set to appear for her own town hall on CNN on Oct. 23, with voters in Pennsylvania. That event will be aired live.

CNN has said that Mr. Trump has an open invitation to appear on the network for a town hall.

Fox News on Friday said the same regarding Ms. Harris. “Fox News has a standing invitation to Vice President Harris for a town hall event of equal stature which has been extended to her campaign multiple times since she became a candidate for president in August,” the network said in a statement.

Mr. Trump, whose performance at the ABC debate in September was widely viewed as lackluster, has repeatedly signaled that he has no intention of meeting Ms. Harris again on the debate stage before Election Day.

Chris LaCivita, one of Mr. Trump’s campaign managers, formally declined Fox News’s invitation for a debate in Pennsylvania on Oct. 23 or 27 several hours after the network first proposed it, according to a person familiar with the exchange. Ms. Harris’s campaign has not commented on the Fox invitation to debate, with the anchors Bret Baier and Martha MacCallum moderating.

If Ms. Harris accepted the Fox News invitation, it could allow her to sharpen her argument that Mr. Trump is afraid to joust with her face-to-face — even on a channel where he enjoys sympathetic coverage from commentators.

Agreeing to a Fox debate, though, could increase pressure on Ms. Harris to appear on Fox for her own solo town hall, which her campaign has so far been reluctant to do.

In an interview set to air next week, Ms. Harris’s husband, Doug Emhoff, told the MSNBC host Joe Scarborough that he believed Mr. Trump was afraid to debate the vice president.

“You saw the first debate, didn’t you? Yeah, that’s why,” Mr. Emhoff said when Mr. Scarborough asked what was stopping Mr. Trump, according to an advance excerpt. “He’s afraid that that’s going to happen again.”