Crawley have spent heavily this summer in 'Project Promotion' as they bid to win the Blue Square Bet Premier.
Red Devils boss Steve Evans was given a major cash injection in the summer by owner Bruce Winfield and used it to bring in Salisbury forward Matt Tubbs, Peterborough winger Sergio Torres and York striker Richard Brodie, who was thought to cost around £250,000 - a record for a non-league club.
But Pires, who has won the World Cup, a European Championship and two Premier League titles, would be by far the biggest coup for the club, who have never played in the Football League.
The veteran, who started his career at Metz and played for Marseille before joining Arsenal 10 years ago, recently said that he would retire if he does not find a new club by the end of January and mooted Aston Villa, Birmingham and QPR as possible destinations.
Evans denied a newspaper report earlier this month that he would have £1m to spend in January.
Crawley visit Newport County in the FA Cup fourth qualifying round this weekend.
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