President Vladimir V. Putin on Monday said American companies could do lucrative business deals in Russia and even help mine rare earths in Russian-occupied Ukraine, further amplifying the Kremlin’s message to President Trump that there was money to be made from a better relationship with Moscow.

Mr. Putin, speaking in an interview with Russian state television that was released late Monday, said that Russia had an “order of magnitude” more of rare earth metals than Ukraine and that Moscow was “ready to work with our foreign partners, including the Americans,” in developing those deposits.

He said the invitation applied to Russian-occupied Ukraine, as well, and added that American companies could also “make good money” by helping to develop aluminum production in Siberia.

“There is something to think about here,” Mr. Putin said, referring to aluminum, “as well as about joint work on rare earth metals and in other areas, for example, energy.”

The interview showed Mr. Putin working hard to appeal to Mr. Trump’s interest in profits and deal-making. It came as Mr. Trump has swung American foreign policy in Mr. Putin’s favor at dizzying speed, as underscored by the United States siding with Russia in votes at the United Nations on Monday about the war in Ukraine.

Mr. Putin praised Mr. Trump’s recent comments about a potential deal in which the United States, Russia and China would halve their defense budgets.