In the House, eight Democrats last week proposed the Paying a Fair Share Act, which would require incomes above $2 million to be taxed at a minimum of 30 percent. That higher tax would start phasing in for incomes above $1 million, and the sponsor of the bill, Rep. David Cicilline (D-R.I.), said this change is estimated to add $53.6 billion in federal tax revenues over the next decade.

"If we want to guarantee fairness in our tax code, then surely middle class families should not be carrying a higher tax burden than millionaires and billionaires," Cicilline said.

Cicilline's bill is co-sponsored by Reps. Matt Cartwright (D-Pa.), Keith Ellison (D-Minn.), Jim Langevin (D-R.I.), Rick LarsenRick LarsenFor Hill staffers, Cruz’s ‘liked’ porn tweet a nightmare scenario Dem lawmaker: ‘Idiocy’ for ESPN to remove announcer Robert Lee from Virginia football game Trump's 'infrastructure week' goes off the rails MORE (D-Wash.), Barbara Lee (D-Calif.), Mark PocanMark PocanDems celebrate Bannon's exit Lawmakers slam Venezuelan leader over 'illegal' election Senate Dems propose crackdown on foreign lobbyists MORE (D-Wis.) and Jan Schakowsky (D-Ill.).

The bill is the House version of legislation introduced by Sen. Sheldon WhitehouseSheldon WhitehouseJuan Williams: Momentum builds against gerrymandering Overnight Regulation: FTC launches probe into Equifax | Dems propose tougher data security rules | NYC aims to slash greenhouse gas emissions | EPA to reconsider Obama coal ash rule Overnight Cybersecurity: Kaspersky to testify before House | US sanctions Iranians over cyberattacks | Equifax reveals flaw that led to hack MORE (D-R.I.). Whitehouse's bill is a part of the Senate Democratic proposal to avoid the sequester, which the Senate will try to pass next week.

The Democratic bills come just as House Republicans appear to be willing to allow the $85 billion in sequester cuts take place on March 1. GOP leaders in the House have said they would only consider legislation to avoid the sequester that is passed by the Senate.

But with the Senate out this week, the upper chamber will have just four days to move legislation when it returns next week. And even if the Senate can pass a bill, it's not clear that the GOP-led House will have any interest in passing a bill that relies on new tax increases to offset the pending cuts.