Mr Biden has previously resisted calls to overhaul or reform the court of nine lifetime-appointed justices, which includes three nominated by Trump.
But that may be changing, according to several US media reports, citing unnamed sources familiar with the planning.
It was unclear whether Mr Biden would come out in support of the measures in the near future or in his second term if he wins re-election in November, The New York Times reported.
However, the measures Mr Biden is reportedly considering – including backing a constitutional amendment that would overturn the court’s ruling on presidential immunity – would require bipartisan congressional support that is almost certainly out of reach.
In addition to the controversy surrounding the rulings, the court has also been engulfed in ethical scandals.
Conservative justices Samuel Alito and Clarence Thomas refused to step aside on cases related to the 2020 election, even as flags linked to Trump’s false election claims were discovered to be flown outside Alito’s home, and Thomas’s wife was part of Trump’s effort to overturn the vote result.