Pressure is mounting on the US Senate to act on gun safety in the wake of the Uvalde and Buffalo massacres, as Republican intransigence continues to stand in the way of all but modest reforms.

On Wednesday the House of Representatives passed a package of gun safety measures designed to staunch the disaster of mass shootings. The extent of Republican resistance was underlined by the fact that only five out of 208 House Republicans voted for the legislation.

Now the focus – and with it the anger of victims’ families and gun safety advocates – turns to the Senate. With the chamber divided 50-50, and 60 votes needed to overcome the filibuster, there is no chance of Democrats passing any changes unless they can bring Republican leaders with them.

As compromise talks continue, there are tentative hopes that a deal might be in the offing. But the outcome is likely to be dictated by Republican refusal to contemplate anything other than limited steps.

The most promising proposals under discussion include plans to increase resources for mental health treatment, school safety provisions and money to encourage states to introduce “red flag laws” that remove guns from those who might harm themselves or others.

On Thursday a group of more than 220 CEOs from some of the biggest brands in the US sent a joint letter to the Senate in which they lamented what they called the “public health crisis” of gun violence. “We urge the Senate to take immediate action … Transcend partisanship and work together to pass bold legislation to address gun violence in our country,” the letter said.

Among the signatories were business leaders of some of the most familiar corporations, including Levi Strauss & Co, Lyft, Unilever US, Yelp and the Philadelphia Eagles.

Chip Bergh, CEO of Levi Strauss, said in a statement that it was time for senators to act. “Inaction on federal legislation has made gun violence a uniquely American problem,” he said.

Gun safety groups are also piling on the pressure on the Senate. Kris Brown, president of the Brady campaign, pointed out that the Senate has sat on legislation to tighten federal background checks on gun sales for the past 15 months.

“That’s 15 months of lethal inaction,” he said.

The most visceral cries for something to be done are coming from relatives of those who died in the recent massacres. Kimberly Rubio, the mother of Lexi, 10, who was killed in last month’s mass shooting in Uvalde elementary school in Texas, told a House committee this week that it was time for a ban on assault rifles of the sort used to murder her daughter.

“We understand for some reason, to some people – to people with money, to people who fund political campaigns – that guns are more important than children. So at this moment, we ask for progress,” she said.

A ban on AR-15 style rifles – the sort used in both Uvalde, where 19 children and two teachers were killed, and in Buffalo, New York, where 10 Black people were killed at a grocery store – is not on the agenda for the Senate compromise talks. The discussions, which are being led by the Democratic senator from Connecticut Chris Murphy and his Republican counterpart John Cornyn from Texas, are focusing on more granular measures that have greater hope of moving forward.

Other ideas on the table are a proposal to introduce juvenile records into federal background checks for anyone under 21 trying to buy a firearm. Senate Republicans will not countenance raising the age limit to buy AR-15s to 21, even though both the Uvalde and alleged Buffalo shooters were aged 18.

Mass shootings continue to be an epidemic in the US, occurring far more frequently than high-profile disasters such as Uvalde and Buffalo. The Gun Violence Archive, which tracks mass shootings, defined as events in which four or more people are shot or killed, counts 251 such incidents in the US so far this year.