The Telegraph says a priority for Hammers chief David Sullivan, as it was in January, is to increase West Ham's attacking options.
Boselli would be an exciting signing. The 24 year-old has one international cap but is tipped to be a surprise inclusion in Diego Maradona's World Cup squad having impressed for his club, Estudiantes, who recently retained the Copa Libertadores.
The striker has scored 23 goals in his last 34 games and has attracted interest from other Premier League clubs, including Birmingham City and Blackburn Rovers although, more recently, Manchester United were also mentioned as a possible suitor.
Estimates on a fee range from £5 million to £1 million with a deal complicated by claims that Boselli's former club, Boca Juniors, are due 50 per cent of his transfer.