The singer, 28, is in the early stages of a relationship with the Hollywood star following her split from music producer Andrew Watt, 27, after 13 months in September.
Garfield, who played the lead in 2012’s The Amazing Spider-Man and the sequel in 2014, romanced La La Land actress Emma Stone, 30, for four years until 2015.
A source close to the singer said: “Rita let slip to pals about her brewing romance about a month ago.
“But in recent days things have started to heat up and she says what they have is the real deal.
“It is still very early days but they like each other and are seeing what happens.
“Their schedules make things difficult but they have a real connection.”
The Oscar-nominated actor, born in Los Angeles and raised in Surrey, was pictured kissing actress Susie Abromeit, 36, in September but they have since separated.
Both Andrew and Rita attended the Evening Standard Theatre Awards separately a fortnight ago.
Tonight was promoting her new album Phoenix in Australia while he was believed to be at home in the UK.
Earlier this year he was seen having dinner with his former Spider-Man co-star Emma despite their separation.
Speaking about her last year, he said: “I’m constantly inspired by her work. I’m constantly inspired by how she handles and holds herself.
“So for me, I’ve — it’s been bliss to be able to watch her success and watch her bloom into the actress that she is.”
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