Timberlake wrote about the conversation with the 15-time major winner in an article for TIME, which named Woods as one of its 100 most influential people of 2019 on Wednesday.
Timberlake said the two spoke about being fathers in a 2018 conversation.
"We discussed how we want our children to see us and how you balance your dedication to them with competitive drive," Timberlake wrote. "I told him about how my son, who just turned 4, has started to realize what I do."
"We bonded over the fact that we wanted our children to watch us at our best. Tiger told me that he hoped his children, who are 11 and 10, would one day see him win a major."
Timberlake, 38, married actress Jessica Biel in 2012. The celebrity couple welcomed son Silas Randall Timberlake in April 2018. Woods, 43, has two children, 11-year-old Sam Alexis and 10-year-old Charlie Axel.
Sam was born almost exactly one year before Woods won his last major -- before Sunday's win -- at the 2008 U.S. Open.
Timberlake said Woods' 2019 Masters Tournament win was a dream come true for Tiger and "all of us."
"Over the past few years, Tiger rehabbed rigorously from all the surgeries, and returned to the top of the game. While pundits doubted he'd ever win again, Tiger's commitment never wavered," Timberlake wrote.
"The thing we don't always appreciate is the patience. Tiger spent countless hours in the gym, and on the golf course, with a singular goal in mind. All the work he did when people weren't watching allowed him to hug his children on the same hallowed ground he hugged his father 22 years ago, amid all the adulation, a Masters champion once again."
The 10-time Grammy-winning musician is a close friend of Woods and they frequently enjoy golf together.
Woods told Bleacher Report in 2018 that Timberlake can consistently shoot rounds in the 70s and has a "beautiful" golf swing.
Timberlake shot a 75 in March at Sherwood Country Club in Ventura County, Calif., according to the USGA's handicap database. He carded a 71 at the same course in November.