The captain has until the end of the week to get on board.
The Herald and Boston Globe both reported that the deadline is Saturday, while The Associated Press reported on Tuesday that Varitek must decide by Friday.
The Red Sox offered Varitek a one-year, $5 million deal to return for his 13th season with the team, the Herald reported. The captain and leader of two World Series champions, Varitek turned down arbitration in December hoping to work out a multiyear deal. But the sides have not been able to agree.
The Herald reported that the Red Sox's offer contains a $5 million team option for 2010 but also allows Varitek to exercise a $3 million player option if the team declines its opportunity.
Varitek had arguably his worst offensive season in the big leagues last year, batting only .220 with 13 homers and 43 RBIs. The switch hitter is a lifetime .263 hitter who has made three All-Star teams and won the Gold Glove in 2005.
The Red Sox reportedly decided on the Jan. 31 deadline to help resolve their catching situation before pitchers and catchers report to spring training on Feb. 12. They signed backup Josh Bard this offseason but may pursue a trade if Varitek does not sign.
"I really don't have anything to say," Scott Boras, Varitek's agent, said after the deadline was disclosed.
The Red Sox have set deadlines with other Boras clients, including pitcher Daisuke Matsuzaka before the 2007 season and free agent Mark Teixeira before the first baseman signed with the New York Yankees last month.
Since Varitek declined the Red Sox's offer to go to arbitration, his desirability to other clubs decreased because the offer of arbitration requires a club to give up a draft choice if it signs the catcher.
Varitek has spent his entire major league career with Boston, which obtained him and pitcher Derek Lowe from Seattle for reliever Heathcliff Slocumb on July 31, 1997.
Information from The Associated Press was used in this report.