Sanofi's board last week authorized the company to press ahead with a formal offer for Genzyme worth up to $70 per share, or $18.7 billion, sources told Reuters at the time.
The two sides are communicating but remain far apart on price, one source said.
A second source familiar with the situation said Genzyme was unlikely to accept a deal under $80 per share, or $21.4 billion.
Genzyme and Sanofi could not be immediately reached for comment.
Shares of Genzyme closed at $70.36 on Nasdaq and traded up 1.5 percent to $71.40 after hours.
Sanofi -- facing the loss of patent protection on key medicines -- is looking to acquisitions to boost earnings and help grow its sales line.
France's Sanofi is seen as the sole buyer for Genzyme and will likely end up paying $74 to $77 a share, or $19.7 billion to $20.5 billion, for the U.S. rare diseases specialist, Citigroup analysts said in a research report.
Citi analysts Mark Dainty and Yaron Werber said $80 a share was unlikely, in part, because Sanofi would be assuming the risk arising from Genzyme's manufacturing problems.
Given that Sanofi is likely the only buyer, and that shareholders are aware that Genzyme could likely trade down to low/mid $50s in case a deal is not reached, we see the deal getting done at $74-77, the analysts said.
Key activist shareholders Relational Investors and Carl Icahn, holding 3.8 percent and 4.9 percent of Genzyme, respectively, could walk away satisfied with an offer in the mid-$70s, since they bought in at $61 and $54, Citi added.
The Wall Street Journal reported that GlaxoSmithKline , Johnson & Johnson and Pfizer have made casual approaches to Genzyme in the past. The suitors asked Genzyme to keep them in mind if it ever considered selling itself.
(Reporting by Toni Clarke, writing by Jessica Hall; Editing by Gary Hill and Tim Dobbyn, Phil Berlowitz)