TMZ reports that Sheen contacted Anderson after she was found in the Two and a Half Men star's Plaza hotel room when he was hospitalised.
According to messages obtained by the website, Anderson sent a text to Sheen, accusing him of destroying her Prada purse in the evening.
Sheen responded: "All I need is an Acct number etc and I will wire u 20k if u think that will cover everything... I really feel bad, u are as cool and sexy and as sweet and fun and friendly as they get! (sic)"
He added: "Don't worry about our mutual friends, deal with me directly and I promise u kind lady, all will be restored and set straight."
Sheen and Capri then exchanged a series of messages, in which the alleged escort told the actor she was "really nervous" about the transaction. It is thought that no cash was ever exchanged.
Earlier today, Sheen filed a lawsuit against Anderson, alleging that the adult film star has tried to extort $1 million (£627,000) from him.