The bill still has to be approved by the New York State Assembly, but that body is dominated by Democrats, making its passage all but certain.
The bill -- the latest round in a battle between the Republican president and the Democrat-led House of Representatives -- will then have to be signed by Democratic Governor Andrew Cuomo, who has already signaled his support for the plan.
Introduced by Democratic-Senator Brad Hoylman, the bill authorizes the state's fiscal authorities to share tax declarations with three congressional committees in Washington.
In most US states, taxpayers have to submit three separate tax returns -- one to the federal government, one to the state and one to the city or municipality they live in.
On Tuesday, the US Treasury refused a demand by Congress to hand over Trump's federal tax returns, which it was demanding as part of the wide-ranging investigation into Russian interference in the 2016 US elections.
The submission of the former real-estate mogul's New York state tax returns could offer a first, if partial, insight into the current financial situation of the president, who has declared himself to be a multi-billionaire.
Trump is the first president since the Watergate scandal in the early 1970s not to make his tax returns public.
Two years ago, the MSNBC journalist Rachel Maddow uncovered Trump's tax filings from 2005, but nothing more recent has so far been unearthed and the president has refused to any effort to release them.
On Tuesday, the New York Times published a report that Trump declared business losses of $1.17 billion between 1985 and 1994.