Sen. John Cornyn (R-Texas), one of the key Republicans to negotiate a bipartisan gun safety package passed last summer, on Monday addressed calls for additional gun legislation in the wake of a school shooting in Tennessee , saying Congress has gone about as far as we can go on measures like background checks.

President Joe Biden pleaded for lawmakers to pass a ban on assault-style weapons and more curbs on firearms after three adults and three children were killed at The Covenant School in Nashville, but Cornyn described his appeals as tired talking points.

I would say weve gone about as far as we can go unless somebody identifies some area that we didnt address, Cornyn told reporters when asked about further legislation and background checks specifically.

During an appearance Tuesday on CNN This Morning, White House press secretary Karine Jean-Pierre said it was devastating for the parents who lost their children, and Americans at large, to hear lawmakers state that the country had run out of options on gun safety measures.

They lost their kids yesterday, and thats what were saying? Jean-Pierre asked. We should not be saying theres nothing else to do. We should be trying to figure out what else there can be to do.

Biden had appealed to lawmakers to make some more progress on reducing gun violence.

Its ripping our communities apart, ripping at the very soul of the nation, he said Monday at the White House, adding that we have to do more to protect our schools.

Congress is unlikely to pass a ban on assault-style weapons, given that Republicans control the House of Representatives and some Democrats have also voiced their opposition to such a measure.

Just [to] be clear, I aint for banning assault weapons, Sen. Jon Tester (D-Mont.) told HuffPost.

He added that itd be great to get something done on background checks.

In June, Biden signed the Bipartisan Safer Communities Act , which included modest curbs on buying guns, as well as funds to boost mental health care and school security. The bill also mandated enhanced background checks for those under the age of 21 and restricted abusive domestic partners from obtaining guns.

He also signed an executive order earlier this month to make more firearm sales require the seller to conduct a background check.

At the time, Biden said the order would move the country as close as we can to universal background checks without new legislation.

But Mondays attack on The Covenant School ignited calls for more action.

Tennessee police fatally shot the assailant, identified as 28-year-old Audrey Hale, on the scene Monday. Hale, authorities said, was in possession of three weapons: a rifle, an assault-style pistol and a handgun.

At least two of those weapons were obtained legally in Nashville, according to police.

Officers reportedly recovered additional firearms when they searched Hales home.

Igor Bobic contributed reporting.