Palace were thought to be in the driving seat to snap up the 36-year-old former England captain, who revealed earlier this week that he would be leaving the club he joined as a youth player in 1995 at the end of the season.
Reports claimed Terry would prefer a move to Selhurst Park because it would mean that he does not have to move home.
However, Eagles boss Allardyce has poured cold water on the speculation purely because he has plenty of defensive options and he appears keen on making Mamadou Sakho’s loan move from Liverpool a permanent arrangement.
“I’ve got Scott Dann, James Tomkins, Mamadou Sakho, Damien Delaney and Martin Kelly, so in central defence we’re looking pretty secure,” he said.
“I’m not sure if that’s going to be a position we’ll look at if and when we’re safe.
“John, fantastic player that he is, would do us a great job, but with that amount of centre-halves at the club, I can’t see that happening at the moment.”
Arsenal secured their place in the quarter-finals of the Europa League following their 3-1 defeat of AC Milan at the Emirates Stadium on Thursday.
Danny Welbeck scored twice against AC Milan to toast his England recall by sending Arsenal into the last eight of the Europa League.
Ipswich striker Joe Garner will not play again this season after it emerged he suffered a fractured skull against Sheffield United last week.