Each eligible person would receive $1,200 stimulus checks, up to $6,000 per household, according to the newly released legislation.
For example, a married couple filing taxes jointly would receive $2,400. They would get an additional $1,200 for each of their dependents, up to three, for a maximum benefit of $6,000.
Under the first round of checks, eligible individuals got a one-time payment of up to $1,200, or $2,400 per couple, plus $500 for each child up to age 16.
Those terms frustrated many progressives, led by Rep. Alexandria Ocasio-Cortez, D-N.Y., by excluding many college-aged kids from ages 17 to 24, as well as undocumented immigrants who paid taxes.
The new round of legislation that's set for a vote in the House on Friday aims to change the "shortcomings" of the initial round of payments, House Democrats say.
College students, non-child dependents and immigrants who pay taxes would also be eligible to receive the $1,200 payment. In addition, families could receive retroactive $500 dependent payments they missed out on last time, according to a House aide.
Undocumented immigrants without a social security number can still file taxes with an Individual Taxpayer Identification Number (ITIN). These ITIN filers could receive the full $1,200 under the Democrats' new plan.
ITIN filers are commonly undocumented immigrants. Legal immigrants working in the United States typically need to get Social Security numbers to get paid, just like American citizens.
The IRS created the ITIN, a tax processing number, in 1996 as a way for individuals not eligible to receive benefits like Social Security to pay federal taxes on the wages they earn.
The benefits of undocumented immigrants filing federal taxes include getting access to economic supports such as the Child Tax Credit (CTC) and the refundable portion known as the additional tax credit (ACTC) as well as showing they are complying with federal laws and have good moral character in the event Congress ever passes a pathway to citizenship.
In 2010, more than 3 million federal tax returns were filed with ITINs, which accounted for more than $870 million in income taxes, according to the National Immigration Law Center. That same year, more than 3 million unauthorized workers paid over $13 billion into Social Security, the center found.
Income caps would be the same as the previous round of legislation under this new bill. Individuals earning up to $75,000 would receive the full benefit and those making up to $99,000 would be eligible for a partial payout. Married couples filing jointly earning up to $150,000 would get the full stimulus amount, while those making up to $198,000 would be eligible for a reduced payment.
The legislation is expected to pass the Democrat-controlled House on Friday, but it faces hurdles in the GOP-led Senate where Republicans want to put a pause on stimulus spending.
Fox News' Adam Shaw and Fox Business Network's Hillary Vaughn contributed to this report.