La Gazzetta dello Sport says today's meetings between the two clubs could be decisive in negotiations.
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Roma are seeking €23m to sell the former Chelsea striker, which falls well within West Ham's transfer budget.
Abraham is also willing to take a cut on his current €6m-a-year wages to smooth the return to London.
Roma are eager for a swift conclusion as they seek to add to coach Daniele de Rossi's squad themselves in the final days of the summer market.