WASHINGTON—Democrats are renewing their push for tougher gun control, seeking to expand background checks and build on the passage of a compromise firearms package last year. 

Sen. Chris Murphy (D., Conn.), a longtime proponent of gun control in the Senate, reintroduced the Background Check Expansion Act on Thursday, a bill that would expand federal background checks to all gun sales and private transfers between individuals. Bipartisan legislation last year expanded checks for buyers under 21 years old. 

“Requiring a background check for every sale or transfer of a firearm would save thousands and thousands of lives,” Mr. Murphy said. 

The reintroduction of the bill follows a shooting at an El Paso, Texas, mall Wednesday night that killed one and left three wounded. The shooting took place steps away from the site of another mass shooting in 2019 that killed 23 people. It also comes days after another shooting on the campus of Michigan State University left three students dead and five wounded. 

Forty-seven members of the Democratic caucus have signed on as co-sponsors of the bill, including Sens. Angus King (I., Maine) and freshman Sen. John Fetterman (D., Pa.). Reps. Brian Fitzpatrick (R., Pa.) and Mike Thompson (D., Calif.) have introduced companion legislation in the House. 

The bill faces headwinds in both the Republican-controlled House, where it is unlikely to get a vote, and the Senate, where 60 votes are needed to advance the bill, requiring bipartisan support. 

Many Republican senators who voted for last year’s Bipartisan Safer Communities Act, a narrow legislative compromise that expanded background checks and increased funding for mental-health and community initiatives, declined to comment Thursday on Mr. Murphy’s bill.

“Not yet,” said Sen. Joni Ernst (R., Iowa), a member of Republican leadership who backed bipartisan legislation last year, when asked if she had initial thoughts on the bill. 

Sen. Marco Rubio (R., Fla.), who voted against the Bipartisan Safer Communities Act, voiced some openness to gun legislation, but voiced his frustration with provisions to expand background checks. 

“The problem is all these people that are doing these things are all passing background checks,” he said Thursday. “Background checks only tell you what they’ve done. They don’t tell you what they’re going to do.”

Some centrists in the Senate Democratic caucus haven’t embraced the proposal. Sen. Jon Tester (D., Mont.) and Sen. Joe Manchin (D., W.Va.), both of whom are facing tough re-election battles in 2024, haven’t signed on to Mr. Murphy’s bill. Sen. Kyrsten Sinema (I., Ariz.), who is also up for re-election, hasn’t backed the bill either. 

Mr. Tester said Thursday he hadn’t seen Mr. Murphy’s bill, but would back “common sense background check bills to keep the guns out of the hands of criminals.” 

Mr. Manchin voiced support for his own proposal. “I’m still in support of Manchin-Toomey, and I always have been,” Mr. Manchin said, referring to a proposal he introduced with former Sen. Pat Toomey (R., Pa.) back in 2013 that would have required background checks for guns purchased on the internet and at gun sales. The Manchin-Toomey amendment has failed to clear the Senate every time it has been introduced.