Blackett made three appearances for Celtic last season
New manager Jaap Stam is keen to bolster the Royals squad before the new season, and the left sided defender apparently fits the bill for what the Dutchman is looking for.
Blackett, now 22, has been capped at every England youth level from under-16s through to the under-21s, and has made 11 appearances for the Red Devils to date.
Theo Walcott hits a cross which is deflected off Manchester United's Tyler Blackett to score an own goal
He was one young defender who Louis van Gaal had turned to due to repeated injury crises at Old Trafford, making all of his United appearances in the 2014-2015 campaign.
Last season he did not have much of a look in, and he was shipped out on loan to Scottish champions Celtic.
But he made a limited impact in Glasgow, appearing only three times in a green and white shirt, and it is unlikely he will return to Parkhead again this summer.
Blackett's unveiling at Celtic
Blackett is likely to have his first-team opportunities at Old Trafford curtailed under the stewardship of José Mourinho, and may once again find himself looking for a new club on a short term basis.
Stam is allegedly hoping to be first in the queue, and he will be hoping to play on his good relations with the club, having played a key role in one of Manchester United’s finest hours in their history: the 1999 treble.
Jaap Stam is keen on Blackett
Stam spent three years at Old Trafford, making 79 league appearances before joining Lazio in 2001.