Amid stories of parents searching in vain for their children, audio of wailing toddlers and images of detained teenagers behind chain-link, cagelike fences, there is intense pressure for Washington to act.

Senators of both parties say they are aiming for a narrow piece of legislation that would effectively codify the executive order that President Trump issued last week, which put a stop to the administration’s practice of removing migrant children from parents who seek asylum.

But while Mr. McConnell said Tuesday that he would like to see the Senate vote before the end of the week, that appears unlikely. After a preliminary negotiating session on Monday evening, both Mr. Cruz and Ms. Feinstein said Tuesday that they do not expect to wrap up their talks until after the July 4 congressional recess.

“I hope we can come to common ground,” Mr. Cruz said. “If we can come together around two core principles — No. 1, keeping families together, keeping children with their parents, and No. 2, enforcing the law and not encouraging illegal immigration — then I believe we will have legislation that can pass both houses of Congress and be enacted into law.”

That is a big if.

Mr. Cruz and Ms. Feinstein are joined in their talks by two colleagues, Senators Thom Tillis, Republican of North Carolina, and Richard J. Durbin of Illinois, the No. 2 Democrat in the Senate. Mr. Durbin, who described the Monday evening talks as “the opening conversation,” said there are “clear differences on a few fundamentals.”