Jenna Bush Hager Sees a Piece of Her Mom Laura Bush in Michelle Obama: 'She Put Her Girls First'

Jenna Bush Hager and Michelle Obama will always have a special connection, formed 14 years ago as the Bushes prepared to hand over the White House to the newly elected first family.

"We've stayed very much in touch throughout these last many, many years," Bush Hager, 40, tells PEOPLE at the Today show studio, moments after interviewing Obama about her new book, The Light We Carry, in front of a live studio audience.

In her on-air conversation with the former first lady, Bush Hager was quick to bring up Malia and Sasha Obama, noting that the book — which tells stories of the daughters navigating adulthood in Los Angeles — made her realize just how grown they are.

"When I first met her girls, they were the age of my children now," Bush Hager says. "They were little kids, and now they're grown adults." Malia turned 24 this past summer, and Sasha, 21.

The Today show co-host says that, thinking about how Obama worked hard to protect her girls from the ugliness of politics, she's reminded of her own mother, former first lady Laura Bush.

"I think [Obama] did for her girls what my mother did for me," Bush Hager says, "which was she put us first."

In 2008, days after Barack Obama was elected the 44th president of the United States, Bush Hager and her twin sister, Barbara Pierce Bush, gave a young Malia and Sasha the full White House tour, sharing fun secrets about the historic home along the way.

Laura had hoped the bonding moment would make the new first lady feel at ease about upending her girls' lives. "My mom knew that she was probably a little hesitant and slightly worried about moving her girls in the middle of the year," Bush Hager says.

What resulted was a beautiful friendship between the families that remains active to this day.

Hoda Kotb, who hosts the Today show alongside Bush Hager, has had a front-row seat to some of their precious moments.

"To show the depth of their relationship and friendship, when Jenna got this job, Mrs. Obama wrote her a note," Kotb tells PEOPLE. "And in the note, it was basically saying that she was so proud of her. Now that, just that statement, think about getting that letter handwritten. 'I'm proud of you.'"

Kotb continues: "[Mrs. Obama] even said to me today, 'We've been cheering her on for so long.' They want Jenna to succeed. And I thought that was the coolest."

Now, as Malia pursues a career in television writing and Sasha studies psychology at USC, Bush Hager says, "We want the same thing for her daughters."