The 26-year-old, a free agent after leaving Swansea this summer, has held talks about a possible move to Tynecastle.
McKay has interest from elsewhere and is currently weighing up his options but Savage is hopeful that the winger will decide to return to Scotland four years after leaving Ibrox to sign for Nottingham Forest.
The sporting director said: "We've had initial conversations with Barrie. He's a very good footballer. We feel he's someone who would improve us.
"We've spoken with his agent and he's got interest from England as well so we've not made any contract offer or anything like that.
"We've just had initial discussions and we're waiting for Barrie to come back to us to see where he's at. He is a player that's been put to us and he's a player we like."
Hearts are also working to tie down John Souttar on a new contract although Savage acknowledges that, with the Scotland defender having made a strong start to the season following a catalogue of injury problems, the club are at risk of losing him for nothing when his current deal expires next summer.
Savage confirmed: "We've offered John a new contract. We're hopeful that he'll be persuaded to stay because he's a top player. He's taking his time to see what his options are. We're fine with that because with the standard and level of player John is, he's bound to attract interest.
"We're trying and hoping to get him to stay but if we don't then we won't be selling him cheap in January, I can tell you that.
"The good thing now is that the window is closed so we don't need to worry about trying to bid for him but it's now up to Robbie and I to get a contract to him that he's happy to sign."
Hearts added Beni Baningime, Alex Cochrane, Ben Woodburn, Cammy Devlin and Taylor Moore to their squad in the summer transfer window, while also offloading several out-of-favour players.
Savage added: "I'm pleased with the business we've done and the quality of player we managed to attract.
"We could probably have got another attacker in although we're still seeing if we can do something on that front.
"A big thing for us was streamlining the squad because it was a bit too heavy before - we had too many players. We've managed to get a lot out and we've got some very good footballers in, so I'm pleased.”