His father has been a senator in the neighbouring state of Vermont since 2007.
The elder Mr Sanders told WMUR, a local television station, that he was “very proud of Levi's commitment to public service and his years of work on behalf of low income and working people.”
“Levi will be running his own campaign, in his own way, with his own ideas,” he continued.
“The decision as to who to vote for will be determined by the people of New Hampshire's first district, and nobody else.”
Bernie Sanders ran a strong challenge during the 2016 primaries against Mrs Clinton, who had long been regarded as the favourite for the Democratic Party's nomination.
Mrs Clinton prevailed but lost to Republican candidate Donald Trump in November's presidential election.
Levi Sanders is not the only member of the family to draw inspiration from Bernie Sanders' political ascent: his step-daughter, Carina Driscoll, has announced plans to run for mayor of the city of Burlington, a post her stepfather once held.
