Federal and congressional investigators are now examining what exactly Mr. Kushner and the Russian banker, Sergey N. Gorkov, wanted from each other. The banker is a close associate of Mr. Putin, but he has not been known to play a diplomatic role for the Russian leader. That has raised questions about why he was meeting with Mr. Kushner at a crucial moment in the presidential transition, according to current and former officials familiar with the investigations.

The New York Times first reported the meeting between Mr. Kushner and Mr. Gorkov in March, but the White House at the time did not explain its aim. That article quoted a White House spokeswoman, Hope Hicks, who said that the meeting came at the request of the Russian ambassador to the United States, Sergey I. Kislyak, with whom Mr. Kushner had met earlier in December at Trump Tower to discuss opening a communications channel with Russian officials during the presidential transition.

But the half-hour meeting with Mr. Gorkov since has come under increasing scrutiny. The current and former American officials now say it may have been part of an effort by Mr. Kushner to establish a direct line to Mr. Putin outside of established diplomatic channels.