The 39-year-old pop star has been in a relationship with the 27-year-old Iranian actor and fitness instructor for more than four years.
The pair met on the set of a music video in 2016 and Spears recently praised him for his support during her "hardest years".
She posted a video on Instagram of herself wearing a diamond ring, writing: "I can't f****** believe it", alongside diamond ring emojis and a heart.
In the video, Asghari asks if she likes the ring and she replies: "Yes!".
Asghari's manager Brandon Cohen confirmed the engagement to People, adding: "The couple made their long-standing relationship official today and are deeply touched by the support, dedication and love expressed to them."
It comes just days after Spears's father Jamie asked a court in Los Angeles to consider terminating the conservatorship that has controlled the singer's personal and financial affairs for more than a decade.
The legal arrangement was put in place in 2008 after she suffered a series of mental breakdowns but she has recently been fighting to end it, accusing her father of abusing his role.
During a hearing in June, she had said the conservatorship was stopping her and Asghari from marrying and starting a family.
Spears has been married twice before. She wed childhood friend Jason Alexander in Las Vegas in 2004 but that marriage was annulled a few days later.
Later that year, she married dancer Kevin Federline, with whom she had two children, before the marriage ended in divorce in 2007.