Reached Wednesday by ESPN.com's Pierre LeBrun, Sundin's agent, J.P. Barry, said his client will make his intentions known "by week's end. Could be tomorrow at the earliest, Friday or the weekend."
The player's camp continues to evaluate both offers. Sundin was skating in Sweden on Wednesday.
Canucks general manager Mike Gillis said he thought Sundin would make a decision by Thursday and was confident he'd wind up in a Vancouver sweater.
"We reiterated our position in all of this and formalized our offer moving forward," Gillis told reporters Tuesday. "We feel good about our team and where we sit and what we have to offer."
Working in the Canucks' favor is that they have plenty of room under the $56.7 million salary cap to sign Sundin. Vancouver has offered him $10 million per year, which would earn him a prorated salary of about $5 million for the rest of the season. Sundin reportedly earned $5.5 million with Toronto last season.
The Rangers are limited in how much salary-cap room they have and would have to make some deals to sign the former Maple Leafs captain. Reports indicate New York would need to free up $2.8 million to sign Sundin for the second half of the season.
Gillis on Tuesday would not say if the Canucks have tweaked their offer.
"We will see how it plays out," Gillis said. "The reality is we've done everything we can possibly do to attract a high-level player to come here. He likes our team, he likes our style of play. He likes the makeup of our team from the goaltender out."
Heading into Tuesday night, the Canucks were first in the Northwest Division with 37 points. The Rangers led the Atlantic Division with 42 points.
The 37-year-old Sundin would bring experience and a scoring touch to whichever team he signs with. The 6-foot-5, 231-pound forward has spent 17 seasons in the NHL with Quebec and Toronto, totaling 555 goals and 766 assists.
A nine-time All-Star, Sundin had 32 goals last season with Toronto. He is the Maple Leafs' career scoring leader with 987 points (420 goals, 567 assists) and served as the club's captain in 10 of his 13 seasons there.
Information from The Associated Press was used in this report.