The Hammers are aiming to close a deal for the England U21 international before Monday night's transfer deadline.
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The Sun says Toulouse will sell for €15m, with West Ham putting together the funds to meet their asking price.
Cresswell was in action for Toulouse against European champions Paris Saint-Germain last night, scoring in the 6-3 defeat.
The youngster began his pro career with Leeds before moving to France.