Both players are out of contract at the Liberty Stadium at the end of next season but Rodgers is keen to tie them down on longer deals before the new campaign gets under way.
Midfielder Britton has established himself as a key performer since returning to South Wales from a short spell at Sheffield United 14 months ago, while winger Sinclair played an influential role in the Swans' promotion to the Premier League and has carried his impressive form into the top-flight.
Rodgers told the South Wales Evening Post: "We are in discussions with Leon at the moment.
"When he came back to the club, he came back on a deal that was about him proving himself again and proving himself to me as a new manager.
"There's no doubt he has done that, and he is one we would like to tie down.
"We have not started anything major yet with Scotty, but I would also hope to get a new contract sorted out with him before the end of the season.
"He and I have had a chat and the chairman has spoken to his agent, so we will see how things go on that one."
Rodgers' focus is on Britton and Sinclair after many of Swansea's key players penned long-term deals following the club's promotion last season.
"A lot of the others were secured last summer, and they are two we would like to add to the list of players who have committed to the club in the long term," he added.
"Leon has become a very important player for us, and part of the contract he signed when he came back was that if he played a certain number of games, his contract would be up for renewal.
"That's where we are at now with Leon and we want to get things sorted out with him.
"Scotty will also be going into the last year of his contract next year and that's not something we would like to run on for much longer.
"He has been brilliant for this club and he is someone we want to secure going forward.
"These things can take a wee bit of time to sort out, but the most important thing is that Scott's very happy here."