Fred Wertheimer is the founder and president of Democracy 21, a nonprofit, nonpartisan organization that seeks to promote government accountability and integrity. Former Ambassador Norman L. Eisen, a CNN contributor, is a senior fellow at The Brookings Institution and served as President Obama's "Ethics Czar" from 2009-11. The opinions expressed in this commentary are theirs.

(CNN) Recent revelations about Cambridge Analytica raise serious questions about whether President Donald Trump's campaign received illegal support from a foreign national. These same questions also exist regarding John Bolton and his super PAC. These matters need to be investigated by the Department of Justice, including special counsel Robert Mueller, as well as by the Federal Election Commission.

Both the Trump campaign and the Bolton Super PAC retained the services of Cambridge Analytica, a data research firm headed by CEO Alexander Nix, a British citizen. The Trump campaign reportedly paid nearly $6 million to Cambridge Analytica over the course of the 2016 presidential campaign, and the Bolton Super PAC reportedly paid Cambridge Analytica $1.1 million since 2014.

According to a New York Times report , in July of that year, an election lawyer for Bracewell & Giuliani advising Cambridge Analytica warned that Nix had to recuse himself and could not be involved in any "substantive management" of any of their clients participating in American elections. The memo from the lawyer also warned about the involvement of other foreign nationals in significant campaign activities. The warning was sent to Nix, then Cambridge Analytica vice president, board member Steve Bannon and board member Rebekah Mercer. (Bannon stepped down from the company in 2016 when he left to work on the Trump campaign.)

Federal law prohibits a foreign national from participating in American elections. An FEC regulation implementing the law says a foreign national shall not "directly or indirectly participate in the decision-making process of" a political committee with regard to the political committee's "election-related activities."

Cambridge Analytica denies Nix was ever in charge of US campaign-related activities and Nix has since been suspended from his job. And former Trump campaign officials said they never relied on Cambridge for voter targeting or persuasion, and that Nix's comments were "an overblown sales pitch."

But if Nix is telling the truth, his work for Trump and his campaign failed to comply with federal law and with the lawyer's advice to Cambridge Analytica that as a foreign citizen Nix could not be involved in managing "the work and decision-making functions, relative to campaign messaging and expenditures."

Thus, it appears the Trump campaign may have received substantial support from a foreign national. Furthermore, the Trump campaign either knew or should have known that receiving such support could be illegal.

The Trump campaign is not the only entity put on the spot by Nix's revelations. John Bolton and the John Bolton Super PAC also may have received similar illegal foreign support from Nix.

According to another New York Times report , Bolton's super PAC was one of the earliest clients of Cambridge Analytica, "which it hired specifically to develop psychological profiles of voters with data harvested from tens of millions of Facebook profiles, according to former Cambridge employees and company documents." According to the report, the relationship between Cambridge and the Bolton PAC grew so close "that the firm was writing up talking points for Mr. Bolton."

A Justice Department investigation must determine whether the John Bolton Super PAC received similar kinds of assistance as it appears the Trump campaign may have -- and if so, whether Bolton knew or should have known that federal law was being violated.

Bolton, through a spokesperson, denies wrongdoing or knowledge of it and a spokesman for the Bolton Super PAC said it "was completely unaware of anything Cambridge Analytica did until recent press reports." And Cambridge Analytica has said the company "adheres to FEC regulations" and that "all CA personnel in strategic roles were US nationals or green card holders and these strategic roles provided all direction to non-strategic personnel."

But general denials won't cut it here.

Bolton needs to answer specific questions, including about his personal knowledge. Did he ever meet with Nix? If so, did Nix ever make any statements to him of the kind revealed on camera last week? Did others at Cambridge Analytica make statements that suggested any foreign guidance of Cambridge's work for the Bolton Super PAC? Was that suggested by the source or content of the materials Bolton received from Cambridge? Did anyone else involved with Cambridge, such as Bannon or the Mercer family, make problematic disclosures to Bolton? (The Mercer family was heavily involved with the Bolton Super PAC in addition to their role in Cambridge Analytica. According to Federal Election Commission filings, Robert Mercer donated $5 million to the super PAC between April 2014 and September 2016.)

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Former Cambridge Analytica employees told the Washington Post that, in 2014, it "assigned dozens of non-U.S. citizens to provide campaign strategy and messaging advice to Republican candidates." We also know that Cambridge Analytica did extensive work for the Bolton Super PAC during that same period, separate from those Republican candidates. This raises the question of whether similar staffing was provided to the Bolton Super PAC and to Bolton -- and whether Bolton or anyone else at his super PAC saw any indications of this.

During the 2013-2014 election cycle, the Bolton Super PAC reported paying $341,025 to Cambridge. The super PAC also reported $3 million in independent expenditures to support Republican Senate and House candidates.

If Bolton knew or should have known that his super PAC received illegal foreign support, that is highly relevant to the new position he will assume next month as national security adviser. One of us (Eisen) has helped vet candidates for this precise position and hundreds of other national security positions. Even benign relationships with foreign nationals are the subject of personnel scrutiny.

If an appointee has benefited from illegal foreign support, this creates the risk of more revelations that could worsen that person's exposure. All of that is, of course, not just relevant to personnel vetting, but to obtaining a security clearance -- even an interim one of the kind this administration is known for.

Moreover, whatever answers Bolton gives, DOJ and special counsel Mueller in particular , should also consider examining the issues regarding the Bolton Super PAC and Cambridge Analytica. Mueller is already examining the connections between the Trump campaign and Cambridge Analytica, as well as foreign interference more generally in the 2016 election. While Russia has been his focus, Mueller has been given broad latitude to pursue matters related to his existing review.

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Since he is already looking at the Trump-Cambridge relationship, he could easily take on the Bolton-Cambridge relationship as well. The facts may overlap, and so may the campaign finance legal issues. Consolidating the matter also makes sense as a matter of efficiency within DOJ.

Outside that body, the FEC has its own separate, independent jurisdiction and should also conduct a review. Because of the notorious dysfunction of that agency, however, we cannot accept the possibility of an FEC investigation alone as a substitute for DOJ's work.

Having both agencies investigate these serious issues is essential to ensuring that foreign interests are not allowed to intrude upon American elections.