Julian Castro, 44, the grandson of a Mexican immigrant who would be the first Hispanic elected president, served as secretary of Housing and Urban Development under former President Barack Obama and has long been viewed as a rising star in the party.
If Democrats take control of one house, the Republicans will struggle to get legislation passed. In the 2010 midterms, during the regime of President Barack Obama, Republicans gained control of the Senate and obstructed legislation moved by Democrats.
The conviction and guilty plea on Tuesday of two former members of Trump's inner circle put a spotlight on the Democratic argument that Republicans have failed to check the excesses of corruption under Trump - but that voters could do it for them.
The proposal comes as the White House and conservative Republicans in the US Congress were edging away from a threat to pick a new budget fight with Democrats after Republicans initially floated some $60 billion in cuts a few weeks ago.
The election, held to replace a Republican who resigned amid a scandal last year, was the latest forceful electoral showing for Democrats, who also won a governor's race in Virginia and scored a U.S. Senate upset in conservative Alabama.