The Russian businessman, who already owns London’s Evening Standard, will reportedly pay a token £1 for control of the papers, in exchange for his promise to invest millions over the next five years.
The final sale hurdle was a 10-year printing deal with publisher Trinity Mirror. Lebedev would initially only commit to five years but that sticking point has now been overcome.
Lebedev has also agreed a five-year deal with INM to supply articles to its other papers in South Africa and Ireland.
The newspaper owner is expected to either slash the £1 daily newspaper price, or make it free - as he has already done with the Evening Standard.
The Independent and the Independent on Sunday currently lose more than £10m a year and last month recorded average circulations of 185,815 and 153,975 respectively.
A source close to the discussions last night told City A.M. that an agreement appears close and that the issue of the printing has been agreed.