The 22-year-old has made only three appearances for the Premiership outfit since his summer switch from Hearts.
Saints manager Tommy Wright said: "He won't just be going for the sake of it, we want the right team for him.
"He's very much part of the squad here but he's not been able to get into the team and we both feel he needs games."
McKay made 13 appearances for Hearts last season but has struggled to make an impression at McDiarmid Park during this campaign.
"He came here from Hearts and he hasn't played as much as he would've liked because of the way others are playing," Wright continued.
"He's been working hard in training but needs to get playing again.
"Brad is still contracted to us for 18 months so he's still very much our player and in our thoughts, but at this moment in time he needs football.
"If the right club comes in we'll look at it, if not he'll stay here and fight for a place."
St Johnstone defender Brad McKay has joined Dunfermline Athletic on loan for the remainder of the season.