A source told the Daily Mirror newspaper the pair are getting "increasingly close" less than a week after her sudden split from 18-year-old Conor Kennedy was revealed.
The 21-year-old 'Lego House' singer recently unveiled his new 'Red' tattoo on his forearm said to be a homage to Taylor's new album, and is identical to the album's font.
Taylor has previously described Ed as one of her "favourite people in the world" and recently revealed that fans can expect more collaborations between them in the future.
She said: "We had both reached out to each other's camps at the same time. I showed up at his hotel in Arizona and we just wrote songs all day.
"Then he came and hung out at my house in Los Angeles and we spent all night sitting on the kitchen floor laughing, writing songs and harmonising.
"We have made other songs and we will put them out at different times."
Ed is currently touring the UK and revealed his plans to wear a costume for his show in Nottingham on Halloween, adding: "I'm gonna dress up in costume for the gig, I hope all of you do too (sic)".