According to The Sun, Scottish giants Rangers are 'quietly confident' they can win the race for the highly-regarded 16-year-old, but they face competition from two English sides.
Weston, who had a £250,000 move to Southampton recently fall through, made his first-team debut for the Seasiders in December of last year.
The youth-team forward came off the bench in Pool's 3-1 FA Cup second round win against Maidstone United.
Born in Liverpool, Weston has consistently impressed for the Under-18s and has won numerous academy player of the year awards throughout his time at Bloomfield Road.
Weston hit 14 goals for Blackpool's youth-team in the early stages of the season and was promoted to first-team training by former manager Simon Grayson in November.
His debut on December 1 capped off a whirlwind week for the youngster, who had to be taken out of school to travel to Suffolk as a member of the Pool squad for the 2-2 draw at Ipswich Town.
The forward was then named as an unused substitute for Pool's EFL Trophy defeat by Scunthorpe United, before making his debut the following Saturday.
Speaking to The Gazette back in December, Grayson said Weston forced his way into the Blackpool team on merit.
“He deserved the opportunity,” Grayson said.
“I’ve not stopped talking about him with my staff since watching him (in a reserve game) at Squires Gate about 10 days ago.
“He went to Ipswich with us last week and he trained with us the back end of the week.
“He’s got so much enthusiasm. The lads were watching what he was doing, where his pure enthusiasm and love of being a professional footballer is at the front of his mind.
“He’s got 30-year-old lads behind him in the warm-up and I’ve never known a 16- year-old lead with stuff like that.
“He’s got confidence in his ability and it’s great for him. He’ll never forget his debut.
“To be fair, he could have had a couple of opportunities, especially on one occasion when Sullay (Kaikai) should have played him in.
“He had good movement and it would have been interesting had the kid put it in, which I think he would have done.”