Speaking in an interview with West Ham TV, the Hammers boss said that he was looking at completing as many as four signings in the coming weeks, with a striker among the top priorities for the London club.
The Irons have already tabled a £12 million bid for Ecuador star Enner Valencia but with the Pachuca striker attracting plenty of interest from clubs both abroad and in the Premier League, the completion of a deal still looks some way off.
West Ham were also linked with a move for Slimani earlier this month, after seeing the 26-year-old star for Algeria at the World Cup with a goal and an assist in the Fennec Foxes 4-2 victory over Korea Republic – the first World Cup match in which an African nation has scored four or more goals.
Since then the Sporting Lisbon forward has been attracting plenty of attention, with Newcastle United and Everton among those linked with a move for the striker.
However that situation may now have changed with Allardyce moving to admit that the club was on the hunt for new players.
"I think there are three or four positions - it depends on the budget," he told the club’s channel
"You would say there is still room for a young central defender, particularly a left-footed one because we do run with three and not four," he added.
"Ideally, you would have four... You would also say we need another left-back to go along with Aaron and I think a front-line player and the possibility of a midfield player as well."
Though not commenting on any move for Slimani, the quotes nevertheless suggest those behind the scenes at West Ham may be willing to spend big this summer.
And if the end result is the arrival of the 26-year-old at Upton Park, it could be money well spent.