The 24-year-old joined Boro last season from Sunderland and made 23 appearances in all competitions.
The 29 times capped Northern Ireland international, who can play in midfield or in defence, has struggled for regular game time, and could be looking to move away from Riverside.
Although the new Boro boss Jonathan Woodgate wants to take a close look at him before making any decision, the club hierarchy are tempted to cash in on the former Manchester United player.
Celtic boss Neil Lennon has now confirmed his interest in the Northern Ireland midfielder. When asked about McNair, the Hoops boss said: “There’s an interest there, yeah.”
According to reports from The Daily Record, the Bhoys are set to lodge a £5m bid for the Middlesbrough midfielder after David Turnbull’s move to Parkhead was called off.
McNair is high up on his shopping list, and the Celtic boss is impressed by his versatility.
That Lennon has confirmed interest in McNair should excite the Celtic fans, as it shows the club’s seriousness in landing him.
Meanwhile, the Bhoys have announced the signing of Christopher Jullien who becomes their first capture of the summer.