To answer the question, let's look back at New England's off season following the 2007 season/prior to 2008.
Randy Moss, Jabar Gaffney, Mike Wright, Tedy Bruschi, Lonie Paxton, Pierre Woods, Ray Ventrone and Wesley Britt all re-signed with the Patriots rather than sign elsewhere as free agents. Kelley Washington returned as well, even after the Pats declined his option. Although the Patriots did not sign any big name free agents off other teams, they did sign at least seven veteran free agents that played elsewhere in 2007: Tank Williams, Jason Webster, Lewis Sanders, Fernando Bryant, Victor Hobson, Sam Aiken and Marcus Pollard.
Similarly the Saints did not seem to be affected by Bountygate and the suspension of Sean Payton. Prior to the 2013 season New Orleans re-signed players like LB Junior Galette and C Brian de la Puente, as well as veteran free agents such as CB Keenan Lewis and TE Ben Watson.
There is nothing to indicate that this controversy will have an adverse effect on New England's ability to sign free agents, whether they be their own or players currently on another club.