Lundqvist, a two-time Vezina Trophy finalist, and the Rangers have agreed to a six-year deal believed to be worth more than $6.5 million a year, according to the New York Daily News.
Sources told ESPN.com on Wednesday that while both sides are close to a deal, it has not been finalized and any announcement of terms is premature.
Lundqvist, 25, would have been a restricted free agent July 1 if a deal had not been worked out.
The Rangers' franchise player, now 24-21-5 with a 2.37 GAA and .906 save percentage, started the season with stunning numbers, allowing more than three goals just once in his first 24 starts. In December, however, he experienced a slump, allowing four goals or more in six of 12 starts.
That slump coincides with when Lundqvist's father, Peter, suffered an aneurysm and had brain surgery in Sweden. Lundqvist, whose twin brother, Joel, plays for the Dallas Stars, has been reluctant to discuss his father's health, and has refused to link his performance with what is going on in his family.
ESPN.com's Scott Burnside contributed to this report.