The 29-year-old appeared set to join the Merseyside outfit after it became clear that the attacking midfielder wanted to leave Fulham.
However, Dempsey moved to Tottenham Hotspur in a £7m deal on deadline day, with the Reds reportedly refusing to up their offer.
It left Brendan Rodgers with just two recognised strikers – Fabio Borini and Luis Suárez – after the club allowed Andy Carroll to move to West Ham on a season-long loan.
However, Ayre backed Liverpool’s transfer policy, despite Rodgers’ side being nine points behind fourth-placed Spurs.
“People may feel they want to criticise the fact we didn’t have an additional striker,” said Ayre.
“But at what price would you sacrifice spending for something that is not in the long-term interests of the club versus what we just did, which was getting a great price on a great player in Daniel Sturridge?
“It may have left us short, but the pieces just did not come together. We’ve come through that period now. I still feel it was the right decision.”
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