The Hammers have been heavily linked with a move for the striker by the likes of the Metro and their need has become more acute since the news that Andy Carroll is likely to be out until at least September.
Slaven Bilic's side looked well short up front against Maltese minnows Birkirkara in their narrow 1-0 Europa League second qualifying round first leg win last week.
The east Londoners are also without key striker Diafra Sakho due to suspension after he was sent off in the previous round.
But while the Hammers have dithered it seems London rivals Arsenal are the latest club, along with Serie A side Fiorentina and Aston Villa, to register an interest in Destro, according to the Express, who cite La Gazzetta dello Sport.
Hope for the Hammers?
There may be hope for the likes of the Hammers and Villa yet, though, as there must be question marks over whether a club of Arsenal's stature and ambition would seriously be interested in a player like Destro?
Yes, at 24 Destro has lots of potential. Yes, he has scored 41 goals in 117 club games so far for the likes of Roma and last season on loan at AC Milan.
He only has eight caps for Italy so far and was the subject of intense transfer speculation last year being heavily linked with moves to West Ham's London rivals Chelsea and Real Madrid.
Loan to rival a damning indictment?
But it could be telling that Roma were willing to loan him out to a rival last season.
Roma, with their own designs on challenging Juventus for the Serie A title, were happy to send the striker to a key competitor which could be seen as a damning indictment.
There is also the theory that Destro's agent is merely doing his job by drumming up interest in his client to try and kickstart his career with a move.
Of the three clubs, Aston Villa would be most desperate for Destro's signature given they have sold Andreas Weimann and Christian Benteke this summer to Derby County and Liverpool respectively.