A tricky winger, Silva has scored 24 goals in 114 career games so far.
But he would cost the Hammers a pretty penny given he still has three years left on his five-year contract at Braga.
West Ham face stiff competition
But West Ham are among a clutch of second tier clubs also said to be weighing up a move.
A report on Abola.pt claimed scouts of several big European clubs were present for Braga's game against Liberec.
If Slaven Bilic is given money to strengthen his injury-hit squad in January, he will face competition from the likes of Benfica, Celtic, Sevilla, Betis, Montpellier and Lorient for his signature.
What do we know about Silva
If he signs for the Hammers they won't want to leave it until the last minute of the window.
That's because his full name is Rafael Alexandre Fernandes Ferreira Silva.
Silva started playing football with Atlético Povoense and F.C. Alverca in Lisbon and when he turned 18 signed for C.D. Feirense.
The winger made his professional debut in 2012 and went on to miss just one game in that entire league campaign totalling more than 3,200 minutes of action to help his team finish 13th in the second division.
It follows West Ham's transfer policy of targeting young, fit players like Aaron Cresswell who can grow in stature and value during their time at the club.
His form for Feirense persuaded SC Braga to tie him to a long-term deal for an undisclosed fee where he has shone since making his Portuguese top flight debut two years ago.
The Hammers have plenty of midfield talent in their squad but will be without instrumental midfielder Dimitri Payet until at least February.