Barcelona's sporting department, led by Deco, has narrowed its list of striker targets down to two names: Martínez and Tresoldi, having spent much of the summer exploring numerous other options without success.
A move for Álvarez was ruled out after Atlético Madrid refused to entertain any offer, while Inter have been similarly firm in rejecting the idea of selling their captain, with any deal for Martínez expected to cost in excess of €100 million even if the Italian club's stance were to soften.
That has left Barcelona increasingly focused on Tresoldi, the 21-year-old Germany Under-21 international who enjoyed an outstanding debut season at Club Brugge, scoring 20 goals across all competitions.
He has continued that form into the new campaign, scoring three goals in his first three official matches for the Belgian champions, a run that has only added to his growing reputation and market value across the continent.
According to Voetbalnieuws, both Manchester United and Chelsea have also made contact over a potential deal, with both English clubs still looking to add further options up front before the window closes.
Club Brugge are said to be willing to sell, with the player himself weighing up the various offers on the table, though a fee in the region of €50 million would likely be required before the Belgian side give serious consideration to a sale.
For Barcelona, Tresoldi represents the most realistic route to a new centre-forward with Álvarez off the table and Martínez looking increasingly unlikely, and his age means any investment would double as a long-term project as well as an immediate reinforcement, even if his early-career profile makes him a less proven option than the club's original targets.
Tresoldi joined Club Brugge from Hannover 96 last summer and quickly established himself as one of the most productive young forwards in Europe, helping the Belgian side to a domestic league and cup double while also making an impression in the Champions League.
Barcelona head coach Hansi Flick is understood to be a firm admirer, having seen the striker up close when Tresoldi scored against Barcelona in last season's competition, a performance that reportedly left a strong impression within the club's coaching staff.
With Robert Lewandowski having left on a free transfer and Ferran Torres closing in on a move to Paris Saint-Germain, Barcelona's need for fresh attacking options has become increasingly urgent as the final days of the transfer window approach.