The deal on offer from Steve Bruce is understood to be for an initial two years with a third year as a option.
Bruce is keen to bolster his squad for next season as Hull prepare for European football in the Europa League and he hopes to win the race for Lescott's signature.
Lescott was not offered a new deal by City when his deal ran out earlier this summer with the England defender dropping down the pecking order at the Etihad Stadium.
Lescott joined City from Everton for a fee of around £22m in August 2009 - and made 92 Premier League appearances, scoring seven goals.
Several Premier League sides, including Aston Villa and West Ham, have expressed interest in the England defender but he has also options abroad, with German side Schalke and Al Ain in the UAE also reported to be keen on his services.