The Chile ace joined Manchester United from Arsenal last month and scored his first goal against Huddersfield on Saturday.
He missed a penalty before scoring the rebound to help United to a 2-0 victory.
But, according to the Sun, he is not satisfied with his fitness levels and has embarked on a new routine to get in better shape.
“He has been doing an extra half an hour after the main session has finished by himself,” a source told the Sun.
“Nobody had asked him to do it but he feels there is an extra sharpness to be had.
“He has got his head down and you can see the steam coming off his body after working outside, it’s so impressive.
“He doesn’t have to do it, he just insists on it. You will not find a better pro around really, he’s superb.”
Sanchez, 29, has already made an impact in his first three games and Jose Mourinho has changed his mind on summer transfers.
Mourinho will no longer look to bring in any more forwards having snared Sanchez from Arsenal.
Sanchez can play in so many roles that United will turn their attention to the midfield and defence this summer.
They are likely to sign a strong midfielder to give Paul Pogba even more licence to raid forward.
Pogba’s performances have dipped in recent weeks and he was dropped on Saturday after a poor showing against Tottenham at Wembley.