Just 17, Shaw has tasted first team football for the Gas.
The Sun says Villa, Leeds and Southampton are all following Shaw's situation.
It's been suggested the Gas will sell for £1m in January.
After their draw with Tottenham in the EFL Trophy last week, Rovers boss Matt Taylor said: "I thought he was the shining light from start to finish. I just thought his performance really stood for something tonight. From start to finish I thought he was our best player so I'm pleased that he's had a good night.
"We are still conscious that he is still only 17 so you do have to manage him a little bit and he's running as hard as he possibly can. Now we want to keep him in and around the first team and see if we can get him some game time and then see how he plays in a Rotherham or Stockport away or something like that."