If a deal is not done by then, then negotiations will be cut off and he will test free agency in the fall, sources told ESPN The Magazine's Buster Olney.
General manager John Mozeliak declined Saturday to characterize negotiations.
The Cardinals exercised a $16 million option for next season on Pujols' contract in October.
Both sides want to keep the negotiations private.
"Really, we don't want to sit here and handicap or guess or kind of give any type of gut feel where this thing is headed right now," Mozeliak said. "Hopefully in the near future it's something we can talk more about, but right now I'm not prepared to."
Mozeliak also declined to discuss negotiations at a fan forum, where he requested the topic not come up, saying, "I have no idea."
Although Cardinals chairman Bill DeWitt expressed confidence earlier this winter that the team could keep Pujols beyond 2011, Mozeliak said: "I don't think it's going to benefit anybody for me to weigh in at this point."
The 30-year-old Pujols led the National League with 42 homers and 118 RBIs last season. He topped 400 career homers and set a franchise record with his 39th multihomer game.
"Obviously, Albert is the face of this organization and the face of this city," pitcher Chris Carpenter said. "He's the best player in baseball. You're talking about huge dollars and huge commitments.
"But my job is to get myself prepared to pitch the best I can and give my team a chance to win."
Carpenter, 35, also can become a free agent after this season. He said he wouldn't worry about it.
"It's been brought up, talked about," Carpenter said. "But I have no concerns, I have no extra pressure. If they want to keep me, they can keep me.
"If they don't, I guess I'll have to get another place to play."
The Cardinals got one deal out of the way, avoiding arbitration and signing setup man Kyle McClellan to a one-year contract. Mozeliak said a long-term deal was discussed but "never gained any traction to this point."
The 26-year-old right-hander appeared in 68 games last season, going 1-4 with a career-best 2.27 ERA and two saves.
McClellan was the only Cardinals player who filed for arbitration. He's been durable in all three of his major league seasons with St. Louis, totaling 202 appearances.
"Obviously, avoiding the hearing room was something that we felt was very important," Mozeliak said. "I'm pleased we were able to work this out.
"Certainly, he's accomplished a lot over the last three years."
Mozeliak expected McClellan to remain in his setup role next season, given the Cardinals have a set rotation of Adam Wainwright, Carpenter, Jake Westbrook, Jaime Garcia and Kyle Lohse. St. Louis added backup options on Friday, signing right-handers Miguel Batista and Ian Snell to minor league contracts with invitations to spring training.
Mozeliak said the Cardinals pursued Batista in 2007 when the pitcher signed with Seattle.
St. Louis also signed 34-year-old middle infielder Ramon Vazquez to a minor league contract and invited him to spring training.
Information from The Associated Press was used in this report.