TSN's Darren Dreger reported that the OK to talk to both men came two days after the Rangers dismissed John Tortorella on Wednesday.
Eakins has been the Marlies' coach since 2009 and has two years left on his three-year American Hockey League contract. The deal includes an out clause that allows him to pursue NHL opportunities this summer.
Eakins coached the Marlies to a 43-23-3-7 record this season before they were eliminated in the Western Conference semifinals. He has received plenty of interest so far from the Rangers and Canucks, as well as for an associate coaching position with the Edmonton Oilers.
Dreger also reported the Dallas Stars have been granted permission to interview Eakins as they look to replace Glen Gulutzan.
Vigneault was fired by Vancouver after the club was swept by the San Jose Sharks in the opening round of the 2013 Stanley Cup Playoffs. He coached the Canucks to the Stanley Cup Final in the 2010-11 season, in addition to six Northwest Division titles and two Presidents' Trophies. He signed a contract extension last year.
Multiple media reports also say Wayne Gretzky and Mark Messier are interested in coaching the Rangers. Messier has no NHL coaching experience; Gretzky coached the Phoenix Coyotes from 2005-09 but the team did not make the playoffs in any of his four seasons.