According to updated numbers provided by the NFLPA following the draft, the Saints have just over $3.1 million in salary cap space following the trade of RB Chris Ivory to the Jets, and prior to the Saints signing any of their draft picks.
The undrafted rookies signed are also unlikely to impact the salary cap figure, as the cap only counts the top 51 contracts on the roster. NFL teams will carry 90 players, a cap that under the new CBA includes their unsigned draft picks, during the offseason into training camp.
Saints GM Mickey Loomis said before the draft that the Saints would have enough money under the cap to sign their draft picks. His plan for that likely accounted for the removal of Ivory's $2.023 one-year contract from the books. That moved the Saints from roughly $1.6 million free to $3.1 million free, as Ivory's deal was in the top 51.
Only five other NFL teams have less salary cap space right now.