The Northern Ireland international is out of contract at the end of the season and is now free to talk to other clubs.
McGinn, 29, has been a key part of the Aberdeen side for four and a half years, helping them secure back-to-back second place finishes over the last two seasons and win the League Cup in 2014.
He is well known to Hearts assistant coach Austin MacPhee who has worked with McGinn in the Northern Ireland set-up.
With the January window now open Hearts are expected to do business under new head coach Ian Cathro and have also been linked with a move for veteran Northern Ireland defender Aaron Hughes.
Hughes, 37, is a free agent after a stint in India with Kerala Blasters. He has played in the English Premier League for Newcastle, Aston Villa and Fulham and is versatile enough to play anywhere in the back four.
Striker Tony Watt, meanwhile, has returned to parent club Charlton Athletic after his season-long loan deal with Hearts was ended early. Watt started brightly but lost his place in the team and finished with just one goal in 17 appearances.