The election interference case against former President Donald J. Trump and 14 of his allies in Georgia has been buffeted by allegations that the main prosecutor, Fani T. Willis, engaged in a romantic relationship with a lawyer she hired to help with the case.

Mr. Trump and one of his co-defendants, Michael Roman, are now trying to convince a judge that the alleged relationship has created a conflict of interest that should be grounds for disqualifying Ms. Willis, the Fulton County district attorney; the hired lawyer, Nathan J. Wade; and Ms. Willis’s entire office.

Though the legal question before the judge will not be taken up until mid-February, some important developments in the matter may unfold this week. Mr. Wade has been ordered to appear at a hearing regarding his divorce case on Wednesday, when he could be forced to take the stand and field questions about his relationship with Ms. Willis. And Ms. Willis is facing a Friday deadline for her legal response to the effort to have her thrown off the election interference case.

The accusations were first made in a Jan. 8 court motion filed by Mr. Roman, a former Trump campaign official. Both Mr. Trump and Mr. Roman are also seeking to have the cases against them dismissed, although these efforts are likely long shots.

The question of whether the prosecutors should be disqualified if the claims are proven true has divided legal experts. Some have called on Mr. Wade and Ms. Willis to resign from the case, in part to avoid the possibility of delaying a trial date.

No matter how the judge rules, the allegations have become a distraction for Ms. Willis, a Democrat and veteran prosecutor who is up for re-election this year. Investigations into her conduct are likely to gear up soon, led by panels controlled by Republicans who enjoy healthy majorities in the Georgia legislature.