The 20-year-old was released by the Magpies in the summer without making a first-team appearance having spent four years in the academy.
Woolston dropped into non-league in the previous two seasons on loan with Darlington, Blyth Spartans and South Shields respectively, although was unable to force himself into the senior picture at St James's Park.
But now the South Shields-born lad is hoping to earn a deal at the Red Devils after being named on the bench for their U23s game with Middlesbrough on Sunday.
A report from the Manchester Evening News said Woolston has impressed the coaches with his distribution.
His most memorable appearance for Newcastle followed in the U23s Premier League International Cup against Sunderland in March.
Woolston, a boyhood Black Cats fan, helped Newcastle to a 11-10 penalty shootout win at the Stadium of Light - even scoring a spot-kick himself.
The goalkeeper admitted after the game: “I am a Sunderland fan,”
“I’ll probably get home and my dad, my brothers and none of my family will speak to me! “I used to be here (at Sunderland) when I was little and got released and came to Newcastle, who have been brilliant with me ever since.
“I always want to win, no matter who it’s against. I might get some stick, but it doesn’t bother me at the end of the day. “I’ve been here about five years and I’ve loved every second.”