Curry officially confirmed that she was leaving the NBC show last week after a number of rumours about her future had circulated.
Us Weekly has now claimed that Lauer was pleased with the decision.
"Matt Lauer is 100 percent behind Ann leaving," a source alleged.
The insider claimed that the duo were "really icy to each other" in the days leading up to Curry's departure, adding: "She refused to speak to him all week."
However, talking when Curry announced her departure, Lauer said that she has the "biggest heart in the business" and told her: "Most importantly, you've made us better, and we thank you from the bottom of our hearts."
Curry will continue to work for NBC as an anchor and correspondent, while Savannah Guthrie has signed up to take her place on Today.