The Trump administration, the aide said, needed to stop sabotaging the law before bipartisan negotiations could begin.

Mr. Trump has established an easy rapport with “Chuck and Nancy,” as he likes to call Mr. Schumer, a fellow New Yorker, and Ms. Pelosi, of California. The three bonded last month over a dinner of honey sesame crispy beef in the White House, and their discussions have gone beyond the fiscal deal and how to protect young immigrants brought to this country illegally from being deported.

They have prodded Mr. Trump to put aside the Republican credo of “repeal and replace” in favor of more modest tweaks to the existing law. They have urged him to preserve subsidies, known as cost-sharing reductions, paid to insurers under the health law to help low-income consumers pay for out-of-pocket health expenses like co-payments and deductibles.

But Mr. Schumer’s quick rebuff of Mr. Trump on Friday shows the limits of the partnership. The senator has said that much will depend on whether the president keeps his promise to protect the young undocumented immigrants who are beneficiaries of the program known as Deferred Action for Childhood Arrivals, or DACA.

“Whether he pivots or not will be one of the most fundamental questions of this administration,” Mr. Schumer said recently. “It’s the $64,000 question. The only way it can happen is if we have a successful negotiation on DACA, and secondly whether we get health care.”

The president’s call is sure to further rankle Republicans, with whom Mr. Trump has had increasingly tense relations over their failure to knock down a key pillar of Mr. Obama’s legacy.

Mr. Trump’s latest move comes at a sensitive moment. The White House is working with Republicans on an ambitious plan to rewrite the tax code, perhaps the party’s last chance for a major legislative victory this year.