Arsene Wenger says Arsenal remain "active" in the transfer market, with plenty of movement expected across the country before Monday's deadline.
Clubs in England and Scotland must complete any remaining deals by 23:00 BST, and Wenger is one manager under pressure to add to his squad.
"We are very active - we are working hard to find a solution," said Wenger.
"Nobody can predict what will happen in the next 24 hours."
Premier League spending has already reached record levels this summer but several clubs still have work to do.
The most recent analysis by financial analysts Deloitte showed that gross transfer spending by Premier League clubs this summer had reached £725m by Friday morning.
Since then QPR have completed the signing of Reading goalkeeper Alex McCarthy, Leicester have bought QPR defender Danny Simpsonand Chelsea have wrapped up a £10.5m deal for Loic Remy, also from QPR, to push that figure still higher.
Arsenal have been linked with a host of strikers since losing Giroud, chief among them Torino and Italy forward Alessio Cerci.
However, any interest Wenger had in the 27-year-old came to an end on Sunday when Spanish champions Atletico Madrid agreed to sign the player on a three-year deal.
Arsenal's need for fresh impetus up front was apparent in their 1-1 draw with newly-promoted Leicester, but Wenger is refusing to be rushed into any kneejerk deals.
"The pressure to buy is always there," he said. "But let's not fool ourselves that the solution every time you don't win is to buy a new player - it would be too easy."
Aside from Wenger's striker search, Manchester City, Manchester United and Liverpool are likely to be involved in last-minute deals.
Any deal that might take Falcao to Etihad Stadium could depend on the future of Alvarao Negredo, who has been strongly linked with a loan move to Valencia.
Liverpool boss Brendan Rodgers, meanwhile, said he might sign "one more" player before the deadline but added that it is more likely that players will leave.
Midfielder Lucas Leiva is among those who could leave Anfield, with ex-Reds boss Rafael Benitez reportedly keen on taking him to Napoli.
The biggest transfer fee paid by a British club so far this summer is Manchester United's capture of Angel Di Maria from Real Madrid for £59.7m, while the busiest top-flight side have been West Brom with 11 new arrivals.
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