WASHINGTON — Democratic leaders in Congress and the White House renewed their push for gun legislation on Thursday, just months after it was defeated in the Senate, amid delicate talks on a new background-check measure that advocates hope could change enough votes from no to yes.

But those negotiations met a warning from Senator Harry Reid of Nevada, the majority leader, who said he would not accept any bill that is substantially weaker than the one defeated in April.

“The bill that passes the Senate must have background checks, and not a watered-down version of background checks,” Mr. Reid declared in the Capitol, flanked by the families of Newtown, Conn., school shooting victims.

Those families also visited President Obama on Thursday at the White House and Speaker John A. Boehner.