The negotiations broke down after they failed to agree on price.
Reports citing sources close to the deal claim the social-buying service prefers an initial public offering (IPO).
The pair were apparently close to closing a £3.8bn deal last week following weeks of negotiations although neither company has officially commented on the reports.
Groupon sends its members details of discounts from merchants which are activated when enough Groupon members agree to purchase.
Reports of the attempted takeover from Google came just weeks after the search engine launched its e-tail venture boutiques.com.
The breakdown in negotiations comes as Amazon confirmed its $175m (£113m) investment Groupon rival LivingSocial.
The investment from Amazon is said to be in return for a 7% stake in the LivingSocial with venture capital company Lightspeed Venture Partners providing a further $7m investment.
This story first appeared on newmediaage.co.uk