Amendments filed over the weekend would revise the bill, so it applies to elected leaders and top public officials like judges, as well as companies in which they have significant ownership or control. The change, expected to be supported by the Democrat-led Senate and Assembly, could ease the way for passage of the overall bill as early as this week.
Under earlier wording, the bill would have applied to any individual or corporation that pays New York taxes.
Democrats say the measure is needed to give congressional committees access to Trump's tax information despite the Republican administration's refusal to hand over his federal returns.
Trump has long filed taxes in New York as a resident of the state. He is the first president since Watergate to decline to make his returns public, often claiming that he would release them if he were not under audit.
Republicans had denounced the bill as an overly broad, partisan attack on the president that could expose all New Yorkers to harassment by Congress.
The bill passed the Senate with its original wording earlier this month and was poised for a vote in the Assembly. Assemblyman David Buchwald, D-Westchester County, said he believes that with the amendment, the measure should pass.
"We recognize that in order to hold top public officials to account for transparency purposes, we could cast a net that didn't necessarily implicate the privacy concerns of all New Yorkers," he told The Associated Press Monday.
The bill is tentatively scheduled for consideration in both the Senate and Assembly on Wednesday.
Gov. Andrew Cuomo, a Democrat, supports the principle behind the bill but will carefully review its details before deciding whether to sign it, according to spokesman Jason Conwall.
Supporters and tax law experts interviewed by the AP had said the privacy concerns were overblown, since Congress already has the power to inspect a taxpayer's federal returns - unless the Treasury Department refuses their request, as they have done with Trump's.
"This power already exists. The issue here is that there's one person on earth - or maybe a handful - who have immunity," said Fordham University Law Professor Jed Shugerman.
The request for tax returns, which would have to come in writing, would only be granted if it is deemed to have a "specified and legitimate" purpose. State tax officials would be directed to redact information such as Social Security numbers that they deem to be private.
All sides expect legal challenges and requests for injunctions if the bill is signed into law, meaning it could be several months before any state tax returns are handed over. The White House did not return message seeking comment on the New York legislation.
"There's no doubt," Buchwald said when asked whether he expects a legal challenge. "The president has always been a litigious person. He often loses those lawsuits. But that doesn't stop the lawsuit."
Republicans in New York said their concerns about the earlier bill language went far behind partisan politics.
"We're going to let them, political hacks, decide when they want to invade your privacy," Sen. Andrew Lanza, R-Staten Island, said during floor debate on the bill. "Today, it's because you are a Democrat, tomorrow it's because you're a Republican, the next day it's because you want to run for Congress."
Lanza said Monday that he will closely review the new version of the bill to ensure New Yorkers are protected.
Congress may not even take advantage of the law if it passes. Under the bill, only the chairs of three specific committees may make a written request for New York returns: Senate Finance, House Ways and Means, and the Joint Committee on Taxation. The leader of one of them, Ways and Means Chairman Richard Neal, D-Massachusetts, has signaled he's not interested.
Neal has issued subpoenas for six years of Trump's tax documents, but Treasury Secretary Steven Mnuchin has so far resisted, saying Congress' request "lacks a legitimate legislative purpose."
But another top Democrat in the House, House Judiciary Chairman Jerrold Nadler, D-New York, has touted the bill as "a workaround to a White House that continues to obstruct and stonewall the legitimate oversight work of Congress."
He noted that the state return would "generally match" the federal return.
Democrats are eager to get ahold of the returns, which could reveal details about his business dealings, his debts and how much of his money came from international sources.
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