BOSTON (AP) — Four of the five candidates in Massachusetts' special U.S. Senate election say they are encouraged by a compromise Senate bill that would subject more firearms buyers to federal background checks and toughen laws against illicit gun trafficking.

U.S. Rep. Stephen Lynch, one of the two Democratic hopefuls, said he would vote for the bill.

"As the Senate moves forward on this debate, I continue to support common sense measures to reduce gun violence," Lynch said in a statement.

Fellow Democratic congressman and Senate candidate Edward Markey said the bill is an important first step but doesn't go far enough.

"We need a comprehensive gun control bill that bans deadly assault weapons and high-capacity magazines while expanding universal background checks," Markey said in a statement.

In a televised debate Wednesday, Republican candidates Gabriel Gomez and Daniel Winslow both said they would vote for the Senate bill.

"I saw the legislation that's being proposed," Gomez said. "It's what I've been talking about from day one — closing the gun show loophole and tie it to mental health and Internet sales."

Winslow said it's important for the country to have a debate on guns. He pressed fellow Republican Michael Sullivan to say whether he would vote for or against the bill.

Sullivan criticized the bill, casting it as a feel-good measure that would do little to keep guns from criminals and the dangerously mentally ill.

"I'm not going to do something just because it feels good," Sullivan said.

Aides to the state's two U.S. Sens. Elizabeth Warren and William "Mo" Cowan, said they back the compromise bill but are pushing for additional measures like re-instituting the federal assault weapons ban.

On Thursday, the bill, which would also increase school safety aid, cleared a key hurdle when the Senate defeated an effort by conservatives to derail the firearms restrictions before debate could begin.

The vote came four months after a gunman killed 20 first-graders and six staffers at Sandy Hook Elementary School in Newtown, spurring President Barack Obama and legislators to attempt to address firearms violence.