One of the sources familiar with the deal said future weapons shipments are still being discussed between Washington and Kyiv.
Trump told reporters that Ukrainian President Volodymyr Zelenskiy wants to come to Washington on Friday to sign a "very big deal."
Officials on both sides have agreed to the draft and advised it should be signed, the sources said.
The deal could open up Ukraine's vast mineral wealth to the U.S.
"What we're doing is now we're saying, look, we want to be secured," Trump said. "The American taxpayer now is going to get their money back, plus.
"Zelenskiy refused to sign an earlier draft of a minerals agreement as Washington sought rights to $500 billion in Ukraine's natural wealth. Kyiv protested it had received far less than that in U.S. aid and the deal lacked the security guarantees Ukraine needs.
Under the terms of a draft minerals agreement, according to sources familiar with its contents, the United States and Ukraine would establish a Reconstruction Investment Fund to collect and reinvest revenues from Ukrainian sources including minerals, hydrocarbons and other extractable materials.
Ukraine would contribute to the fund 50% of the revenue minus operating expenses and continue until the contributions reach the sum of $500 billion. The United States would provide a long-term financial commitment to the development of a "stable and economically prosperous Ukraine."
Asked what Ukraine would get in return for the minerals deal, Trump cited what he said was $350 billion already provided by the U.S. "and lots of ... military equipment and the right to fight on."
Ukraine has deposits of 22 of the 34 minerals identified by the European Union as critical, according to Ukrainian data. They include industrial and construction materials, ferroalloy, precious and non-ferrous metals, and some rare earth elements.
Ukraine's reserves of graphite, a key component in electric vehicle batteries and nuclear reactors, represent 20% of global resources.