The comedian and Spice Girls singer have been on a number of dates, according to The Sun, and were recently photographed together while taking a guided tour of Hampton Court Palace.
Onlooker Greg Oram, who was visiting Hampton Court with his family on the same day, is quoted as saying: "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 he was with a Spice Girl.
"They seemed like they were having a really great time, holding hands and chatting - it was 100% obvious they were a couple. Geri was holding his hand the same way I hold my wife's. It was so cute."
A source close to Brand claimed that "things are getting very serious" between the Rock of Ages actor and Halliwell, saying that they "really got to know each other" during rehearsals for the Olympics closing ceremony and that Brand "is spending so much more time in London just to be with her".
Halliwell ended her relationship with Henry Beckwith last year, while Brand was recently linked with model Isabella Brewster and was even reported to be considering moving in with her.
Photo gallery - London 2012 Olympics - The closing ceremony:
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