A deal was agreed in principle on Wednesday afternoon for the highly-rated 21-year-old centre back.
Diop, who captains the France U23 side, made 37 appearances for Toulouse last season.
The club finished 18th in Ligue 1 last term but secured their top-flight status with a play-off win over Ajaccio.
Sky Sports News understands West Ham are also in advanced talks to sign Swansea goalkeeper Lukasz Fabianski.
West Ham close to agreeing Issa Diop deal with Toulouse
13 June 2018
West Ham are close to agreeing a deal worth up to €25m (£22m) to sign Issa Diop from Toulouse, according to Sky sources.