The 27-year-old singer-songwriter has been renting a property in London while she's been spending time with her new boyfriend.
Swift has gone to impressive lengths to keep their romance quiet.
A source told The Sun newspaper: "Taylor has flown in via private jets and her security has made it a military-like mission to prevent her from being seen.
"She's been walking around with Joe in London using disguises, like scarves and hats, to keep her identity under wraps.
"Nobody has a clue they've been walked past a music superstar - not even her new neighbours."
Swift - who has previously dated the likes of Harry Styles, Calvin Harris and Tom Hiddleston - is eager to ensure her new relationship doesn't descend into a circus, as has previously happened.
Like Swift, the 26-year-old actor is also keen for the duo to keep their relationship low-key, although an insider has revealed the blonde beauty has been open about her new love with friends and family.
The source explained: "This isn't a new couple alert or a secret relationship - the only people this has been a secret to is the media because all of us, Taylor and Joe's closest friends and their families, were aware they have been together for several months and have known each other for a longtime.
"After her 1989 tour and all the attention Taylor received, she learned she had to be more protective of her personal life and she and Joe decided early on to keep their private life private."
- Bang! Showbiz
