The 38-year-old coach has been vocal about his desire to remain with the Swans, with news of a fresh contract emerging last week.
"We're pretty much there," Rodgers told BBC Sport. "I had a good chat with the chairman [Huw Jenkins] a few days ago and we met up again yesterday.
"Once the paperwork is done then hopefully over the course of the weekend that will be agreed and everything signed then you'll have nothing to ask me.
"I'm very honoured and proud to lead such a marvellous club into what is an exciting era in the club's history."
The Northern Irishman is currently on a one-year rolling contract.