The allegations against Sen. Robert Menendez may have started with unsubstantiated stories about trysts with prostitutes, but questions about the lawmaker’s conduct have reached the point where they’re not focused on sex.

A slow drip of national news reports about the New Jersey Democrat and his ties to Florida ophthalmologist and political donor Salomon Melgen have put his office on the defensive over an assortment of issues including a port contract in the Dominican Republic and a possible intervention in a Medicare billing dispute involving the doctor.

Menendez tried to move forward with his normal agenda Thursday, holding a morning roundtable discussion with Hispanic media outlets about immigration through the Senate Democratic Hispanic Task Force, of which he is chairman.

Even there, however, Menendez was pulled off-topic and denied he exerted improper influence to help Melgen in the Medicare dispute.

“The bottom line is we raised concerns with [Centers for Medicare and Medicaid Services] over policy and over ambiguities that are difficult for medical providers to understand and to seek a clarification of that,” Menendez said, according to ABC News and Univision.

Still, the story is creating fodder for Republican campaign operatives; both the outgoing and incoming spokesmen for the National Republican Senatorial Committee have mentioned the Menendez matter on Twitter in recent days.

According to The Washington Post, Menendez brought up a government inquiry into Melgen’s billing practices with senior officials at the Centers for Medicare and Medicaid Services, including the agency’s acting administrator. That raises the question of whether Menendez went beyond normal constituent service.

“People ask our office to intervene on their behalf every day. Some are friends, and some are strangers. Some may be contributors but most are not. Some of their arguments have merit for the policy process to consider,” a Menendez aide said in a statement. “In each case, we take a look at the request, and if we feel any action is appropriate, we take it.”

The New York Times last week reported Melgen also was part-owner of ICSSI, a Caribbean port-security firm, and that Menendez intervened on the company’s behalf regarding a contract worth as much as $500 million. It also reported that Melgen’s company had donated $700,000 to a super PAC that provided $582,500 to Mendendez’s 2012 re-election effort.

Questions of impropriety have lingered despite recurring doubts about charges Menendez spent time with prostitutes during trips to the Dominican Republic, based in large part on the fact that the person originally raising the accusations has never surfaced.

Though the optics of the relationship between Menendez and Melgen are proving problematic for the senator, it is unclear whether any ethics or criminal case could be built on such interactions.

Menendez’s chief of staff sent an email to some of his boss’s closest supporters Wednesday that supplies talking points to combat news reports about the travel and ethics scandal. It was part of a broader public relations offensive to bolster Menendez’s fortunes.

The memo, apparently sent from a personal email address of Chief of Staff Danny O’Brien, encourages Menendez supporters to downplay two private jet rides the senator took two years ago and reimbursed last month. Menendez has acknowledged that he traveled on an aircraft belonging to Melgen.

Sen. Johnny Isakson, R-Ga., ranking member of the Ethics Committee, told The Washington Post the panel was “aware” of the media reports. The Senate’s code of conduct and the ethics rules prepared by the committee during the last Congress state that lawmakers “may not receive compensation from any source because of improper influence exerted from their position as a Member, officer, or employee of the Senate.” Nor can they “solicit or receive compensation for representing another person or entity before a government agency in a manner in which the U.S. government has an interest.”

Asked about the matter Thursday, Isakson said he could not comment and referred CQ Roll Call to prior statements.

In a criminal case, the evidence needed to prove a quid pro quo scheme would be stringent — and the long-standing friendship between the two men would further complicate matters. Sen. Frank R. Lautenberg, a fellow New Jersey Democrat, also declined to comment on the substance of the allegations Thursday.

Bribery cases are notoriously hard to build. The Supreme Court several years ago said the “honest services” statute often used by prosecutors in corruption cases was unconstitutionally vague. The late Sen. Ted Stevens, R-Alaska, was prosecuted for filing false financial disclosure forms, not for engaging in a quid pro quo scheme. Even so, the case later fell apart after it became clear prosecutors had not shared evidence with the defense, and Stevens’ verdict was vacated.

Legal experts told CQ Roll Call it wouldn’t be enough for two seemingly related events to occur one after the other — there would have to be evidence of intent.

“If you don’t have someone wearing a wire, you’re going to have to do it by having people who were close to them tell you the story, and they may or may not be credible,” Peter Zeidenberg of DLA Piper said. “It’s a very tough standard, and [these cases] are hard to prove, particularly when you’ve got a relationship.”

Kate Ackley contributed to this report.