Independent News and Media (INM) is in exclusive negotiations with Lebedev, who bought the Evening Standard in 2008.
Although the deal is unlikely to go through today, it is understood that the two parties will extend the negotiation period to 26 February.
According to The Guardian, INM is keen to push a deal through by 24 March at the latest. The company's results are due that day and a sale of the two loss-making titles would be a bright spot.
It is believed that talks between the two parties hit a stumbling block because of issues revolving around the two newspapers' print contracts, which are handled by Trinity Mirror, as well as deficits and redundancy costs.