The team said in an unattributed statement that the move gives the sides more time to negotiate an agreement. If no deal is reached by July 15, the Patriots will pay Gostkowski $4.126 million in 2015.
Gostkowski led the NFL in total points for the fourth time in 2014. He is the leading scorer in franchise history.
The 2006 fourth-round draft pick was 35 of 37 on field goals this season - a 94.6 percentage that is the highest in Patriots history.
He did not miss a kick in the postseason, converting the only field goal and all 15 extra point attempts while helping the Patriots win their fourth Super Bowl since 2001.
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