Mike Garafalo of the Newark Star-Ledger reports that $4 million is expected to be paid up front.
Weatherford, 29, is a workout maniac who had 82 punts last season with an average of 45.7 yards per punt. He placed 25 punts inside the 20-yard line, with a long punt of 62 yards.
But his most memorable moment of the season came after holding a field goal for Lawrence Tynes. After Tynes split the uprights to defeat the San Francisco 49ers in overtime, Weatherford took his helmet off, sprinted around the field and shouted, "We're going to the [expletive] Super Bowl!"