The Bhoys are in need of defensive reinforcements despite landing Kolo Toure this weekend.
The Bhoys bolstered their back line with the free transfer signing of Kolo Toure on Sunday, however, manager Brendan Rodgers told the Scotsman that Wednesday's Champions League qualifier away to Astana would likely come too soon for the Ivorian.
The Scottish champions can still add one new face to their European squad under the wildcard ruling should Toure fail to make it - provided they arrive before the end of Tuesday.
And despite Celtic's need for defensive reinforcements, it is unlikely the spot will go to McNair, with the Daily Star reporting his future at United will be decided next month.
The Northern Irishman was named in Jose Mourinho's squad to tour Asia this summer, with the Red Devils due to face Manchester City in China on Monday, but for severe weather conditions.
However, Mourinho could still opt to send McNair on loan, with his progress having stalled last season.
The 21-year-old, who occupy centre-back, full-back and central midfield roles, made eight appearances in an injury-hit 2015-16 campaign - down eight on the previous year.
The report claims a number of Championship club's are also said to be keen on securing his services.