"We see this as a serious matter, I think. And we believe the senator stepping down from his chairmanship was obviously the right thing to do," spokesperson Karine Jean-Pierre told reporters.
"As it relates to anything else, any decision that he has to make, that's certainly going to be up to him and the Senate leadership to decide. But of course, we see this as a serious matter, and I'm just going to leave it there for now," she added.
Earlier Monday, Menendez steadfastly rejected calls from fellow Democrats to step down from Congress after he was indicted for a second time last week.
Addressing reporters in his native New Jersey, Menendez maintained his innocence, saying "prosecutors get it wrong sometimes," and stressed that "the court of public opinion is no substitute for our revered justice system."
"I firmly believe that with all the facts are presented, not only will I be exonerated, but I still will be New Jersey's senior senator," he said.
"We cannot set aside the presumption of innocence for political expediency when the harm is irrevocable. For those who have rushed to judgment you have done so based on a limited set of facts framed by the prosecution to be as salacious as possible," added Menendez.
Menendez has faced a growing litany of calls for his resignation from the federal legislature, including from his fellow Democrats. He temporarily stepped aside from his chairmanship of the powerful Foreign Relations Committee Friday after the charges were announced.
A federal grand jury returned a three-count indictment charging Menendez, his wife, and three others with conspiracy to commit bribery, and conspiracy to commit honest services fraud. The senator and his wife are additionally charged with extortion.
The indictment is the second against Menendez during his decades-long career in public service.