The Catalan club pursued Martinez, 23, earlier this summer but Inter’s €111m valuation (his release clause) proved a sticking point.
Lautaro is still their number one target but a deal remains very complicated.
The player’s agents plan to travel to Milan this week, coinciding with Lautaro’s return to training with Inter.
His representatives will meet with Inter to analyse the situation and try and unlock a transfer to Barcelona.
Argentine international Martinez has already agreed personal terms with Barca, but the clubs remain far apart.
Inter and Barcelona remain in regular contact, but there are ‘great economic differences’ between them.
Barca want to pay a maximum of €65m plus a player (possibly Junior Firpo), while Inter are demanding a minimum of €85m.
With time running out, the clubs will again negotiate the potential transfer of Lautaro in the days ahead.
City want Lautaro, but…
Pep Guardiola’s Manchester City have reiterated their interest in Lautaro, who would be a long-term Sergio Aguero replacement.
But for now, Lautaro is only considering leaving for Barcelona or staying at Inter with a significant pay rise.
Ronald Koeman has pitched the name of Memphis Depay to Barca chiefs and the Dutch winger appears to be the leading alternative to Lautaro.
There have been reports in recent days that Depay, 25, is close to joining Barcelona in a €25m deal.
If final negotiations fail for Lautaro, then Barca will more than likely move quickly to secure Depay from Lyon.