The couple have been together for five years, and a friend of British actor Joe (31) said he had bought singer Taylor a "beautiful" ring, which she wears only at home.
The couple, who have kept their relationship private, are thought to have let friends and family know.
The pop sensation (32) and the actor have been engaged for a several months, friends have said.
A source told a newspaper: “Taylor and Joe are incredibly happy, and very, very in love.
“They’ve actually been engaged for a few months but have only told their inner-inner circle — basically immediate family, and trusted, very old friends. Everyone has been sworn to secrecy, too.
“Taylor has a beautiful ring but she only wears it when she’s at home — ie behind closed doors. Again, only a handful of people know details about the wedding and Taylor hasn’t even told some of her team about the engagement.
“They want their love to stay away from the cameras as much as possible. This is just for them. And if and when they do exchange vows, there most definitely won’t be any Vogue, Rolling Stone or Hello! magazines there. It will be simple and elegant — like them.”
The Shake It Off singer lives with Alwyn in north London.
Swift, who is worth €550million, has had eight No1 hits. Alwyn has acted in films including The Favourite, and more recently the BBC series Conversations with Friends, the adaptation of Sally Rooney’s debut novel directed by Irish filmmaker Lenny Abrahamson, the TV follow-up to smash hit Normal People.
In January 2020 Swift revealed her mother Andrea had been diagnosed with brain cancer. The star splits her time between London and Nashville to care for her.