Aaron Wilson of The Baltimore Sun first tweeted news of Costa's visit.
The visit with Costa highlights one aspect of the Patriots' behind-the-scenes free-agent approach as they have had discussions with some depth options along the interior of the offensive line. With starting center Ryan Wendell an unrestricted free agent, the Patriots are exploring some contingency plans.
The 26-year-old Costa started all 16 games in 2011, but has been limited to three games apiece in the 2012 and 2013 seasons because of back and ankle injuries. Similar to Wendell, Costa is on the smaller side for a center (6-foot-3, 302) and would likely project more to a backup role.
Costa was cut by the Cowboys late last week in a move to help the team get under the salary cap.