Curry took over as co-host when Meredith Vieira left the show last year, but has been talking to network officials for weeks about a different job.
The Times said falling ratings, a lack of chemistry with co-host Matt Lauer and Curry's suitability for hard news are being cited as reasons for the change.
Discussions about Curry's future with the network are expected to be finished before NBC's coverage of the Summer Olympics begins in late July, the Times said.
Although no replacement has been hired, Savannah Guthrie, who fills in when Curry is off, has been mentioned, the newspaper said.
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