The free agent striker, available after his exit from Galatasaray, remains open to a return to Italian football having previously held talks with Juventus in January of last year, when Luciano Spalletti gave his approval to a deal and Giorgio Chiellini pushed hard for the move.
Icardi is not yet convinced that Lazio represent the best project available to him, and rival suitor Fiorentina, who deny any interest, have yet to make contact with his camp.
Lazio are not the only club in the running. Sources close to the player say several clubs outside Italy have also asked for information, and Icardi is expected to make a decision in the coming days. Should a significant offer arrive from elsewhere, Serie A's chances of seeing him again would diminish sharply, though it remains possible Lazio could still find an agreement.
Time is not necessarily on Lazio's side. The club needs a swift resolution with the season already under way, and a defeat to Mantova has done little to ease the pressure on a squad already dealing with financial constraints, including issues around its liquidity index, and a support base showing signs of frustration at the Stadio Olimpico.
Away from the Icardi pursuit, Lazio have also identified a younger target in Hutchinson, an 18-year-old England Under-19 international at Leicester City who impressed in the Premier League Under-18s last season.
English reports indicate Lazio have made an approach for the winger, though the Rome club faces competition from other suitors for his signature.
The contrast between the two pursuits underlines the range of Lazio's summer approach: a high-profile, low-risk move for an experienced striker available on a free transfer, alongside a speculative bet on a highly rated teenager still developing in England's youth system.
Lazio sources say they will continue monitoring the market until the closing stages of the window, with Icardi's decision likely to shape how much room remains for further additions.