The teenage forward has been the standout player at the FIFA under-20 World Cup in Turkey, with scouts from across Europe in attendance.
And Portuguese newspaper A Bola claims that Jose Mourinho wants to wrap up a deal as soon as possible before rival bids drive up his price further.
Bruma has hit five goals in two matches for the Portuguese under-20 team in Turkey, and currently leads the competition's scoring charts heading into the knock-out round.
The 18-year-old has been compared to Cristiano Ronaldo, another former-Sporting youth team star, and Bruma could make the same transition from Portugal to Premier League.
But Sporting are desperate to hold on to their prize asset, and are keen to offer the teenager a new and improved contract to keep him at the club.
They privately concede that Bruma will eventually leave, but want a new contract in place to ensure they receive the maximum compensation.
There's also a feeling that once Bruma fully breaks into the first-team next season, the exposure will drive his price higher while also offering him the best chance to develop quickly.
Chelsea have one of the best youth teams in England, having appeared in three FA Youth Cup finals in the past four years, but Bruma will want first-team football after breaking into the main Sporting squad earlier this year.