The 23-year-old singer and the 37-year-old actor have been spotted on a number of romantic dinner dates recently in Los Angeles, but the pair have further fallen for each other since filming together on the set on 'The Giver' in South Africa.
A source told The Sun newspaper: ''Before they left for Cape Town they had a couple of dinners in LA.
''He's much older than the guys Taylor has dated in the past year but maybe that's where she's going wrong.''
Meanwhile, the 'We Are Never Ever Getting Back Together' hitmaker has been seeking love advice from her good friend Ed Sheeran to try and figure out where she has gone wrong with previous failed relationships, with several stars, including One Direction hunk Harry Styles, John Mayer and Jake Gyllenhaal.
She said previously: ''He says there are different kinds of love: There's physical attraction, mental attraction, and emotional attraction; there's also comfort and obsession.
''You need to have all of those things in one person. You have to mentally respect them, be physically attracted, and have a comfort level.
''You're obsessed with them, yet you also know they're going to stay. I've never had that in one person.''
Update: Since this news article was written, a rep has been in contact and issued this official statement: 'Taylor never met Alexander before she arrived in South Africa, and they are not dating.'
She's swapped the studio for a film set with new Netflix series.
Ed Sheeran performed a spectacular set on the Pyramid Stage on Sunday.
Viral video ties in 'GoT' soundbites to the lyrics of Gloria Gaynor classic.
Here are a few of the best moments over the Glastonbury weekend.
The wrestling series hit the streaming service earlier this week.
The Marvel Studios head was blunt in his answer.
Yet another teen sci-fi adventure, this movie may be sharply well-made but it struggles to...
Author Lois Lowry talks about her dystopian sci-fi book 'The Giver' in a featurette ahead...
Dr Seuss' eco-fable is transformed into a raucous adventure comedy in this colourful animated feature....
Ted is a young boy who lives in the perfect town: everything is clean and...
Watch the trailer for Valentine's Day *Chick Flick Alert* Valentine's Day is a romcom set,...
Watch the trailer for Hannah Montana The MovieHannah Montana has had the feature film treatment....