Weems is due a $1 million base salary and has a $1.6 million salary-cap figure in 2014 in the final year of his three-year, $4.2 million contract signed in 2012 that included $1.5 million in guaranteed money.
Weems' agent, Chris Martin, didn't return telephone calls.
The Bears cut wide receiver Earl Bennett earlier this week when he refused to take a paycut.
Weems has a $100,000 workout bonus, $250,000 in incentives through a playoff win plus making the Pro Bowl. He has also up to $500,000 receptions escalator.
Weems was a 2010 Pro Bowl special-teams selection for the Atlanta Falcons when he had 40 kick returns for 1,100 yards and a touchdown and had 18 punt returns for 230 yards and one touchdown with 16 special-teams tackles.
He played in every game last season for the Bears and caught one pass for eight yeards, returned five kickoffs for 57 yards and had nine special-teams tackles and one forced fumble.
In 2012, Weems had 13 kickoff returns for 231 yards and two catches for 27 yards with eight special-teams tackles.
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Aaron Wilson covers the Ravens for The Baltimore Sun