The Hammers splashed the cash on striker Andy Carroll at the start of the summer but saw Carlton Cole, Marouane Chamakh and Wellington Paulista all leave at the end of last season, which leaves just Modibo Maiga as Carroll’s competition.
Arnautovic is just one of a number of strikers Allardyce is looking with Salomon Kalou, Peter Odemwingie and Emmanuel Emineke also linked.
The Hammers boss has nearly emptied to clubs summer transfer budget and so may opt for Arnautovic who is thought to be the cheapest of the four options.
The 6ft3in forward has shown promise in Germany and has scored seven goals in 18 games for Austria but West Ham face competition for their man with clubs in Spain already in talks with Bremen for the striker who would prefer a move to London.
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