Allardyce is anxious to get a deal wrapped up before Meite begins his World Cup campaign with the Ivory Coast and has tabled a cut-price offer.
Meite, who can play anywhere in the back four, is ready to agree terms at around £1million a year to go to The Reebok.
Bolton want to move fast as Middlesbrough and even Arsenal have shown interest in him in the past.
Meanwhile, Wanderers are also eyeing Guinean international defender Daouda Jabi.
Jabi was with France's Ajaccio last season and was watched by Bolton boss Sam Allardyce.