"Hillary, get on with your life and give it another try in three years!" the president posted to Twitter.
Looking past a network like MSNBC, which all but serves as the campaign arm of the Democratic Party, she denounced Fox News for its partisanship.
Also on Friday in another media interview, Clinton lashed out at Trump and Alabama Republican Senate nominee Roy Moore for not taking responsibilities for sexual assault allegations made against them.
The president has already expressed on other occasions his negative views of Clinton competing with him in the elections for the White House, although the former secretary of state has said that "his career has come to an end".
On WABC radio, Clinton said the Franken situation differs from Moore because the Minnesota senator apologized said he would "gladly cooperate" with an ethics investigation. She said Russia, Trump and the GOP were the sources of those things.
"I don't hear that from Roy Moore or Donald Trump", she added.
During the 2016 election campaign, several women claimed that the then Republican presidential candidate had "sexually abused them", an act that Trump has repeatedly denied.
Questioning the legitimacy of the results, she said, "I think it was one of the major contributors to the outcome".More news: Jerry Jones apologizes for inappropriate racial comment made in 2013