Reports on Wednesday say the 30-year-old is a top target for the Hammers if they can offload some fringe players from the wage bill.
Allardyce has made no secret of the fact that he wants to sign a proven striker to partner Andy Carroll when he returns to full fitness and sees the experienced Cardozo as an ideal purchase.
The Paraguay international has apparently told Benfica he wants out and is keen on a move to the Premier League with Upton Park a good destination for him.
Turkish side Fenerbahce, Russian giants Spartak Moscow and even Premier League new boys Cardiff City have been linked with Cardozo, who has an impressive 105 league goals in 160 appearances in the Portuguese top flight for Benfica since he joined them in 2007.
Tottenham's Champions League dream was ended by a second-half sucker-punch from Italian champions Juventus.
Sky Bet Championship leaders Wolves returned to form with a bang as they beat Leeds 3-0 at Elland Road to move a step closer to promotion.
A magical performance from Jordan Jones saw Kilmarnock climb to fifth in the Ladbrokes Premiership with a 2-0 win over 10-man St Johnstone.
Florent Hadergjonaj believes home advantage could prove crucial when Huddersfield play host to Swansea on Saturday.