No formal announcement about the changing-of-the-guard has been made, but the report said the network has committed to making Fallon, 38, the successor to Leno, 62, on "Tonight" and plans to put its plan in motion soon.
The show started in New York in 1954 with Steve Allen as host and moved to California in 1972 with longtime emcee Johnny Carson. Leno took over the show when Carson retired in 1992.
The Times quoted a senior network executive, who did not wish to be named, as saying of the Fallon-Leno switch and "Tonight" move to New York, "There is no way on earth that this is not going to happen."
NBC has even begun work on a new "Tonight Show" studio in its headquarters building at 30 Rockefeller Plaza in Manhattan, the newspaper said.
NBC previously booted Leno as host of "Tonight" at the end of the 2008-09 season, despite high ratings, and gave the position to "Late Night" host Conan O'Brien to keep O'Brien from moving to another network and courting younger viewers in the same time slot.
NBC then offered Leno a nightly, hourlong, 10 p.m. series, which kicked off in September 2009.
But when the prime time "Jay Leno Show" failed to be a strong lead-in for local news broadcasts, NBC announced it would push Leno's show back to 11:35 p.m., shorten it to 30 minutes and air "Tonight" at 12:05 a.m.
However, O'Brien said he would not remain on "Tonight" if it was bumped to 12:05 a.m. and left the network, paving the way for Leno's return as "Tonight" host in the traditional 11:35 p.m. time slot in March 2010.
O'Brien now heads up his own chat show "Conan" on TBS.