WASHINGTON — Talks intensified Tuesday toward a potential end to the government shutdown, as the federal closure was on track to become the longest ever, disrupting the lives of millions of Americans.

But tensions also rose as senators from both parties, Republicans and Democrats, quietly negotiated the contours of an emerging deal. With a nod from their leadership, the senators are seeking a way to reopen the government, put the normal federal funding process back on track and devise some sort of resolution for expiring health insurance subsidies that are spiking premium costs from coast to coast.

“Enough is enough,” said Senate Majority Leader John Thune, the South Dakota Republican, as he opened the deadlocked chamber.

On day 35 of the federal government shutdown, the record for the

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