The 35-year-old kicker hit the free-agent market last month but said a return to New England was his "first choice."
The Patriots originally selected Gostkowski in the fourth round of the 2006 NFL Draft out of Memphis. He is a four-time Pro Bowl selection and three-time All-Pro choice.
Gostkowski helped New England win three Super Bowls during his time with the franchise. He has totaled 1,743 career regular-season points with an 87.4 field goal percentage.
In 16 regular-season contests last season, Gostkowski made 27-of-32 kicks (84.4 percent) with a long of 52 yards. He connected on 49-of-50 extra point tries.