The French star, who would cost £700,000, is currently on £40,000 a week at Roma, but Bolton boss Sam Allardyce has tabled him a contract worth £32,000-a-week.
If Dacourt accepts the offer he will become Allardyce's 3rd signing of the summer, following Ali Al-Habsi from Lyn and El-Hadji Diouf from European Champions Liverpool.