The pair fell for one another while rehearsing for the London Olympics closing ceremony earlier this month.
England's The Sun newspaper has quoted a friend of Russell's as saying: "Things are getting very serious between them - they're a proper couple now. They had met several times in the past but really got to know each other during the Olympics.
"People thought Russell was joking at the time when he said he had a crush on Geri all over again, but it was true.
"They just clicked - they have so much in common. He is spending so much time in London just to be with her."
Geri is a 40-year-old single mum to six-year-old Bluebell and is in the midst of planning a Spice Girls reunion tour but has made plenty of time for Russell including a recent romantic date at Hampton Court Palace in Middlesex.
Fellow visitor Greg Oram said: "I thought it was the most random thing ever to see Russell Brand of all people in a place like that - until my wife pointed out that he was with a Spice Girl.
"They seemed like they were having a really great time, holding hands and chatting - it was 100 per cent obvious they were a couple.
"Geri was holding his hand the same way I hold my wife's, it was so cute."
The love life of Russell Brand has been filling tabloid newspapers since the break up with wife Katy Perry in December 2011.
Photos: Getty Images
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