Reds boss Jurgen Klopp is keen to bolster his forward options, and reports in Portugal reveal he has pinpointed the 22-year-old as a potential recruit.
Liverpool have been pursuing Roma winger Mohamed Salah but fear being priced out of a move, with the Serie A side having placed a hefty price tag on the Egyptian.
The Reds are tracking a number of attacking targets as they prepare for their return to the Champions League next season.
There is definite interest in Bayern Munich's Brazil international Douglas Costa while highly-rated Senegal winger Keita Balde of Lazio is also being monitored.
Martins, though, has emerged as a significant target, having come through the ranks at Sporting and made his international debut for Portugal last October.
Martins, who can play on both flanks but predominantly on the right wing, scored seven goals and contributed 14 assists in all competitions this season.
Sporting value the player at £50million but are likely to sell for less given the financial pressure to sell having not made next season's Champions League group stages.
Barcelona, Real Madrid and Manchester United have also been recently linked with Martins, who caught the eye during a Champions League clash with Real last year.