Blackett, who was handed his debut last season by Louis van Gaal amidst an Old Trafford injury crisis, made 11 appearances for the Red Devils last season.
The England Under-21 international impressed – mainly as part of a three-man central defence – but found himself down the pecking order this season, but United remain keen to hand him playing time.
Having recently been knocked out of the Champions League qualifying stages, Scottish champions Celtic were eager to add reinforcements to their squad, with the 21-year-old top of their list.
The Daily Mail report that the deal could involve a £4million ‘first refusal’ buy-out clause for Ronny Deila’s side.
A centre-half by trade but also adept at left-back, Blackett has enjoyed loan spells at Championship sides Blackpool and Birmingham in the past.
With Blackett’s arrival at Celtic Park, the future of Virgil van Dijk remains in doubt.
A target for Southampton throughout the summer, the dominating 24-year-old centre-half has been linked with a £10million move to St Mary’s recently, and Blackett could well be considered his replacement.