While a source in New York cautioned the agreement could still fall apart, French newspaper Le Monde claimed the parties had agreed on a payment of $6 million (£3.75m).
Le Monde reported that Mr Strauss-Kahn and the maid, Nafissatou Diallo, would meet a judge in New York on December 7 to sign the deal and close an affair that ended the Frenchman's International Monetary Fund career and wrecked his presidential ambitions.
"The discussions have been going on for weeks, months. The agreement should be confirmed at the start of next week," said Michele Saban, a friend of Mr Strauss-Kahn.
"We are moving towards the end of a tragedy," she said, adding that Ms Diallo had always been open to negotiating a settlement.
Le Monde reported that Mr Strauss-Kahn planned to take out a bank loan for $3 million and would be lent the other $3 million by his wife Anne Sinclair, despite the fact the couple separated in the summer.
News of the deal comes as Mr Strauss-Kahn awaits a decision by a French court on December 19 on whether to call off a sex-offence inquiry involving parties attended by prostitutes, where he risks trial on a charge of "aggravated pimping".
Ms Diallo alleged that Mr Strauss-Kahn forced her to perform oral sex on May 14, 2011, in his suite at the Manhattan Sofitel.
The criminal prosecution fell apart after doubts emerged concerning Ms Diallo's credibility as a witness, and the attempted-rape charges against Mr Strauss-Kahn were eventually dropped.
Mr Strauss-Kahn has maintained the sexual encounter was consensual, although he admitted in a TV interview that he regretted his "moral error".
He filed his own countersuit against the maid earlier this year, claiming her accusations had destroyed his career and harmed his reputation.