The couple are believed to have entered a "cooling-down period" due to Brand's reluctance to commit to a relationship so soon after his failed marriage to Katy Perry, which came to an end in December.
"Geri was smitten but things have petered out a tad," a source told The Mirror.
"He isn't ready to settle down so soon after coming out of marriage, whereas Geri is probably more ready to have a relationship.
"She wants a pad and spent Saturday dragging Russell around properties near Devon. It probably seemed a little much."
The publication also claims that even Halliwell's loved ones have warned her not to get too serious with the comedian.
"Russell is a huge flirt and Geri won't stand for any nonsense," the insider continued. "Her friends are telling her not to get serious about him."
Halliwell recently described her relationship with Brand as a "whirlwind" and admitted to being "choosy" with her partners.
Photo gallery - Geri Halliwell's love life:
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