Citing sources familiar with the discussions, All Things Digital reports that that the deal could be done as soon as Wednesday, though its completion is not yet assured.
Google declined to comment, citing a policy of not commenting on rumor or speculation.
If Google does manage to buy Groupon, it will be the company's largest deal to date. Google has made two other acquisitions at a cost greater than $1 billion: DoubleClick in 2007 for $3 billion and YouTube in 2006 for $1.65 billion.
Google has another large acquisition pending, a proposed $700 million deal for travel technology company ITA Software. Google's competitors have been been lobbying hard get regulators to block this deal.
Groupon's valuation in April was $1.35 billion, according to Forbes.
Group buying sites like Groupon have become increasingly popular and numerous, according to Hitwise, an online metrics company. Web traffic to the 81 sites in this category was up more than fourfold for the week ending August 21, 2010, compared to the same period in 2009. Groupon received about half of the visitor traffic during this period.
Facebook's entry into the online coupon market, however, has changed things, perhaps enough to motivate Groupon to accept an offer from Google. At the beginning of November, the social networking giant introduced Deals, a way for businesses with Facebook Place pages to offer online coupons to local customers. The move has been widely characterized as a potential threat to startups like Groupon, Foursquare, and Gowalla -- the fear is that Facebook could simply replicate the startups' business models on its platform.
Ned May, VP of analysis firm Outsell, argues that buying Groupon may not pay off for Google as immediately or as well as the company might wish. He says that local advertising is not so much a technical problem as a people problem.
"While there are synergies in building out a display business and a coupon business, these are not core strengths of Google," he said in an e-mailed statement. "Buying their way in is likely the only way they'll get there, though success afterward is in no way guaranteed."