Curry's status on Today is very much in question, with some reports suggesting that she may be removed from the anchor chair later this week.
Morales was cited as an early contender to replace Curry, but Savannah Guthrie and Hoda Kotb have recently been tipped as the most likely contenders to join Matt Lauer as co-anchor of Today.
An insider told Page Six: "If Natalie's passed over for the job, she's going to leave next… This will be the second time [Natalie] has been passed over for the top job. And she's not happy."
An NBC spokesperson described the claim as "totally untrue" and insisted that Morales "couldn't be happier" in her current role as Today's news anchor.
Morales has been with Today for six years.
It was reported on Tuesday (June 26) that Regis Philbin is currently in negotiations to reunite with Kathie Lee Gifford as hosts of Today's fourth hour if Kotb is chosen as Curry's replacement.
Curry is expected to stay with NBC News after her Today departure.