By Ali Zaslav and Daniella Diaz, CNN

(CNN) -- Democrats are reintroducing legislation to expand background checks on all commercial gun sales, the first congressional move on significant gun control since Democrats won the White House and the majority in both chambers of Congress.

The bill, called the Background Check Expansion Act, would expand background checks for all firearm sales or transfers in the country. Currently, background checks are not required for gun sales and transfers by unlicensed and private sellers.

The issue was a major Democratic priority that President Joe Biden campaigned on in the 2020 election.

The likelihood of actually passing the bill remains a longshot, as Democrats hold a slim 50-50 majority in the Senate and would need significant Republican support to overcome a legislative filibuster. Congress has struggled for years to pass any meaningful gun safety reforms. But universal background checks have been one of the few plans that have actually garnered some bipartisan support.

"Obviously this isn't going to be easy and we're gonna need the White House's help to get it across the finish line, but I truly believe this is the year to get it done," Democratic Sen. Chris Murphy, who is sponsoring the Senate version of the plan, told CNN in a phone interview on Monday.

And Democratic Rep. Mike Thompson, who is reintroducing the House version of this legislation, told CNN in an interview he hopes this legislation would actually reach Biden's desk.

"I'm hopeful. Sen. Murphy, I'm working closely with him. He's working real hard over (in the Senate)," Thompson said.

Thompson noted of the high level of support for this kind of reform. According to a 2019 Pew Research Center survey, 93% of Democrats and 82% of Republicans are in favor of creating background checks for private gun sales and sales at gun shows.

He said the House will vote on his bill before April 1, which is the last day bills can move to the floor without a markup or hearing. The bill also has three Republican cosponsors in the House — Reps. Fred Upton of Michigan, Brian Fitzpatrick of Pennsylvania and Chris Smith of New Jersey.

This is the second time the House would vote on this specific bill. It was first introduced in 2019 and passed the House but was blocked by Senate Republicans, who had the majority in the upper chamber at the time.

Thompson, who is the chair of the Gun Violence Prevention Task Force in the House, said he plans to meet with the White House in the "near future" to discuss this legislation.

"It's a high bar over in the Senate, but I feel good about where we are and I feel good about my partner over there, Mr. Murphy," Thompson said.

The announcement comes two weeks after President Joe Biden called on Congress to take immediate action on common sense gun legislation.

Sen. Pat Toomey, Republican from Pennsylvania who is retiring in 2022, spoke bleakly about the chances of getting enough of his GOP colleagues on board to support reprising his plan to expand background checks with Sen. Joe Manchin, a moderate Democrat from West Virginia.

"It's theoretically possible, but I'm not aware of a significant change in heart," Toomey told CNN last week. The Manchin-Toomey background checks plan failed to pass twice in recent years. Aside from Toomey, the only other Republican to vote for plan that is still serving in the Senate is Sen. Susan Collins of Maine.

But Murphy -- who has been a vocal advocate for stricter gun control since the 2012 Sandy Hook Elementary School shooting that left 20 children and six adults dead in his state -- is hopeful that support has shifted in the Senate since the last time they voted on expanding background checks seven years ago.

"I've fielded phone calls, I've had conversations with senators over the past few years, which you know, it's clear that some people's minds have changed," he said of his Republican colleagues, guessing that if a strong background check bill came to the floor today, there's a "chance" it could get the 60 votes needed to beat a filibuster.

Toomey also noted there are new members of Congress as well since the last time the Senate had this debate, so "we'll see."

Sen. Richard Blumenthal, the other Connecticut Democrat who is also a leading voice on stricter gun control, told CNN last week he's more "hopeful" about gun legislation getting bipartisan support as the movement for gun violence prevention has "skyrocketed" and the opposition, namely the NRA, has "imploded."

"More Americans than ever before support common sense steps and it's now a political movement -- it's no longer scattered pockets of support," said Blumenthal, a co-sponsor of the bill.

The bill is also being cosponsored by Majority Leader Chuck Schumer, who has been a strong advocate on stricter gun control and universal background checks throughout his career, Majority Whip Dick Durbin and Sen. Catherine Cortez Masto of Nevada.

Murphy said he hasn't spoken to Schumer about a when the bill will be brought to the floor, as "there are some pretty pressing issues before the Senate right now." Senate priorities include the $1.9 trillion Covid relief package and Biden's infrastructure plan.

With road bumps ahead in the Senate, many Democrats are hopeful Biden will use his executive authority to enact gun safety reforms that he's championed, like defining "ghost guns" to be within the definition of firearms. They also hope he will close several background check loopholes, such as the "Charleston Loophole," which allows some licensed gun sales to go through before a required background check is done. Using this loophole, Dylann Roof was able to legally purchase a firearm to kill nine people at a historically black church in Charleston, South Carolina, in 2015.