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.
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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.