The Commission on Presidential Debates (“CPD”) is “eligible under federal law” to conduct presidential debates. That means they thread the needle between maintaining their non-profit status (not acting on behalf of any political campaign) and not providing an in-kind contribution to a political candidate. 11 CFR § 110.13 covers some of the rules for hosting a debate. Disputes most commonly arise over eligibility standards (often, independent or minor party candidates litigate whether the CPD’s threshold is permissible, as those candidates are usually excluded). But the CFR includes some rules for the debate format:

(b) Debate structure. The structure of debates staged in accordance with this section and 11 CFR 114.4(f) is left to the discretion of the staging organizations(s), provided that:

(1) Such debates include at least two candidates; and

(2) The staging organization(s) does not structure the debates to promote or advance one candidate over another.

Advisory Opinion 1986-37 offers some insight on this standard in rejecting the form of one proposed debate:

Each invitee will be allowed to speak to the assembled attendees at the convention for 20 minutes on a topic or topics of his or her selection. At the end of this address, there will be a 15-minute question and answer session with questions being asked by attendees from the convention floor. There will be a moderator who will not comment on the questions or otherwise make comments that imply approval or disapproval of any of these invitees.

[] Each invitee will deliver his or her address from the podium on the dais on the convention floor. Specific portions of the convention agenda will be set aside for these addresses so that each invitee will be given a comparable time for his or her address, such as 10 a.m., 2 p.m., or 4 p.m.

…

The Commission notes that your proposed candidate debate features individual appearances by each of the candidates at separate times over the course of the convention rather than concurrent, face-to-face appearances. Such face-to-face appearances or confrontations have historically been an inherent characteristic of candidate debates since the prototypical Lincoln-Douglas Debates in 1858 and, more recently, the presidential debates in 1960, 1976, 1980, and 1984. Although the format and structure of these debates varied from one instance to another, the common element in all of them was a face-to-face confrontation. The Commission's nonpartisan candidate debate regulations were drafted with this historical, traditional concept of candidate debates in mind. Accordingly, the Commission does not view your proposed candidate appearances as constituting a candidate debate.

Another statement from the FEC in 1995 on a related issue:

Nevertheless, the requirement of including two candidates would be satisfied, for example, if two candidates were invited and accepted, but one was unable to reach the debate site due to bad weather conditions, and the staging organization held the debate with only the other candidate present. Other situations will be addressed on a case-by-case basis. The Commission does not intend to penalize staging organizations for going forward with debates when circumstances beyond their control result in only one candidate being present and it is not feasible to reschedule.

The standards, then, are quite flexible—two candidates, not promoting one over another. That said, there is an expectation that they will be “face-to-face” or “confront[]” one another.

As the CPD considers modifying its debate standards for President Donald Trump and former Vice President Joe Biden, there will need to be some confrontation between the two candidates. But it has broad flexibility to work within those parameters—at least, under existing interpretations of the law.