Bush had signed the offer last week, giving the Saints until Tuesday to either match that deal or let Bush go. If the Saints had lost Bush, they would not have received any compensation because they tendered him a minimum offer of $1.4 million for next season.
Bush's new deal is reportedly worth up to $4.5 million over two seasons.
Bush is projected to be a reserve, but see regular playing time after appearing in 13 games last season with six starts.
Bush was credited with 37 total tackles in 2013, along with a fumble recovery and five passes defended. He also made three special teams stops.
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