WASHINGTON (Reuters) - U.S. senators pushed for a compromise gun control bill on Tuesday, a day after the Senate failed to advance four gun measures following last week’s mass shooting in Orlando, the deadliest in modern U.S. history.   Senate Majority Leader Mitch McConnell said he would schedule a vote on a bill by fellow Republican Senator Susan Collins that would prevent about 109,000 people on “no-fly” and other surveillance lists from purchasing guns. Collins said she expected a vote on the bill this week or next. On Monday, the Senate defeated a Democratic prohibition on gun sales to people on a broader range of government watch lists, while also blocking a narrower Republican measure.  Some senior Republicans would not commit to supporting the Collins bill, citing worries about people being denied the ability to buy weapons without adequate safeguards. But the No. 3 Senate Democrat, Chuck Schumer, told reporters: “There may be a glimmer of hope now,” adding that Collins’ proposal seemed to be “a step in the right direction.” The measure is being debated in the Senate before the Nov. 8 election when Democrats hope to win control of the Senate and gain seats in the House of Representatives. Democrats believe Republican congressional opposition to wider control bills gives them a powerful campaign issue. It was too soon to tell if President Barack Obama would support the Collins bill. Spokesman Josh Earnest said senior officials including lawyers at the Department of Justice were taking a look at it. “If the assessment is that this would enhance the ability of our law enforcement professionals to keep us safe and prevent suspected terrorists from purchasing a gun, then that’s likely something that we’ll be able to support,” Earnest told reporters in a daily briefing.   In the Senate on Monday, four measures to expand background checks on gun buyers and curb gun sales to those on terrorism watch lists - two put forth by Democrats and two by Republicans - fell short of the 60 votes needed for passage in the 100-member chamber. The votes were a bitter setback to advocates who have failed to get even modest gun curbs through Congress despite repeated mass shootings. The bills lost in largely party-line votes that showed the political power in Congress of gun rights defenders and the National Rifle Association. The White House accused U.S. senators of a “shameful display of cowardice” and said they failed the American people by not advancing any gun control measures after the Florida shooting. The gunman, Omar Mateen, pledged allegiance to Islamic State during the June 12 rampage in which he killed 49 people and wounded 53 at an Orlando gay nightclub before being fatally shot by police. Senator John Cornyn of Texas, the No. 2 Senate Republican, said he had concerns about the Collins measure. “It’s a slippery slope when an American citizen is denied a constitutional right, without forcing the government to come forward with some evidence on the front end” that a person should be prohibited from buying guns, he said.   After Monday’s votes, the Senate also debated a different tactic for battling domestic attacks that could be inspired by Islamic State and other foreign militant groups. Senators were aiming to vote by Wednesday on legislation by Republican John McCain of Arizona expanding the Federal Bureau of Investigation’s ability to conduct secret surveillance in counter terrorism investigations. “This week we’ll have the opportunity to strengthen our ability to combat lone wolf terrorists and connect the dots so we are better able to prevent terrorist attacks in the United States” such as the Orlando massacre, McConnell said. Collins, who held a news conference with eight other senators including Democrat Bill Nelson of Florida, said her bill would stop about 2,700 Americans and 106,300 foreign national on surveillance lists from buying guns. “We believe that if you are too dangerous to fly on an airplane you are too dangerous to buy a gun,” the Maine lawmaker said. She said Americans and immigrants with work permits could appeal a denial and recover lawyers’ fees if they prevailed.  Nelson said he owed it to the families of the victims in the Orlando shooting as well as police and other first responders to the carnage.  It was not clear whether Collins’ plan would draw significant bipartisan support. The NRA said her proposal was “unconstitutional” and would not have prevented the Orlando attack.  Gun control groups promised to intensify their push for legislative action, and not just in Washington. Shannon Watts, founder of Moms Demand Action for Gun Sense in America, said her group, besides supporting pro-gun control candidates for Congress, would work to strengthen gun-sale background check laws “state by state until Congress acts.” “If the NRA and their lapdogs in the Senate thought moms would feel dispirited and back down, they are sorely mistaken,” Watts told reporters in a teleconference. 