Genzyme Corp. and French drug maker Sanofi-Aventis are in talks over a possible sale, several news outlets are reporting.

The news comes after weeks of speculation, prompted first by comments from Sanofi CEO Chris Viehbacher that it was in the market to buy a company worth about $20 billion. The Sanofi board has reportedly authorized a sale up to $18.7 billion – news that prompted shares of the company to bump the market cap of the Cambridge-based biotechnology company up to just under that sales price last week.  On Monday, some reports indicated that the price of Genzyme could go even higher, to over $20 billion.

Sanofi has now reportedly sent a private letter to Genzyme, detailing its offer of $69 per share, or $18.4 billion, according to several news organizations, quoting unnamed sources. Shares of Genzyme (Nasdaq:Genz) closed at $70.36 on Monday, and Genzyme is reportedly looking for a price closer to $80 per share, according to Reuters.

Genzyme officials were not immediately available for comment today.

Genzyme has reportedly been courted in recent weeks by other pharmaceutical giants, including Merck and Co., Johnson & Johnson, and GlaxoSmithKline. The company may use the existence of potential competitors as a bargaining chip to push the price offered by Sanofi higher.

Continuing manufacturing problems at Genzyme – serious enough create shortages of three drugs and prompt the U.S. Food and Drug Administration to level a major enforcement action against the company  – have been a significant drain on Genzyme’s earnings however. The company’s weakened financial performance of late and the burden of long-term oversight by the FDA during the coming years could provide Sanofi some leverage in negotiations.