The striker has been linked with a Premier League move recently.
The Hammers are thought to have been alerted to the possibility of stealing in for the striker after issues arose between Monaco and Roma over the specifics of the transfer fee.
However, CalcioMercato’s latest report claims the clubs will resume talks in the coming days with Monaco giving a ‘firm offer of £8.6 million’, while Roma want closer to £10.7 million before sanctioning the move.
West Ham will hope the disparity between the fees will allowing them to move in, but it is unclear whether they will be willing to meet the Italian club’s demands.
Destro managed just eight goals in 31 league appearances last term - spending the second half of last season on loan at AC Milan – and doesn’t appear to have a future at Roma.
At 24-years old, there is still plenty of time for the striker to improve and he has shown his quality in the Serie A over the past three years.
A move to the Premier League, however, is a different proposition, and given West Ham's relatively well-stocked frontline, other areas are possibly a bigger priority for Slaven Bilic this summer.