Any such plan could face staunch opposition from the majority congressional Republicans, who would have to approve it. On Thursday, House Speaker Paul Ryan voiced opposition to getting rid of the debt ceiling.
The deal was described as a "gentleman's agreement," the Post reported, citing "three people familiar with the decision." Democrats in the Senate want to reach an "arrangement with Trump" by December, according to the Post.
Speaking to reporters Thursday, Trump did not explicitly say that he and Schumer agreed to push forward on scrapping the debt ceiling. However, he said "certainly that's something we'll discuss" and did talk about at Wednesday's White House meeting with congressional leaders.
Congress has had to repeatedly raise the federal borrowing limit to avoid risking default on U.S. debt. The contentious deadlines can set the stage for political maneuvering and deals on unrelated measures. GOP lawmakers often seek spending cuts to go along with votes to raise the debt limit.
Lawmakers face their latest deadline to lift the debt limit by the end of September. On Wednesday, Trump, Schumer and House Minority Leader Nancy Pelosi struck a deal to attach Hurricane Harvey recovery funding to measures to temporarily extend the debt ceiling and fund the government for only three months.
The president did so over the objections of Republican congressional leaders and Treasury Secretary Steven Mnuchin, who wanted a longer debt ceiling fix. Some GOP lawmakers have slammed the agreement, saying it gives Democrats leverage over budget discussions in December.
Mnuchin later voiced support for Trump's plan.