Sinclair has been strongly linked with a move to South Wales in recent days and could be heading to the Liberty Stadium within the next 48 hours.
Premiership new boys Blackpool are still thought to be very interested, along with Ipswich Town, but Sinclair's links with former Chelsea coach Rodgers is likely make Swansea the more favourable choice.
It's thought that the pacy winger, who spent time on loan at Wigan Athletic last season, is keen on making a permanent exit from Stamford Bridge, despite playing a big part in Chelsea's pre-season campaign.
Today's Sun is reporting that the fee is worth around £1.5m, which would break Swansea's transfer record paid for Craig Beattie a year ago by some margin.
"We've spoken to Chelsea and we hope to tie something up very soon with Scott," Swans boss Rodgers told BBC Sport.
"There's one or two things that we're looking at. This is a squad that isn't going to need major surgery as long as I keep the squad I want to keep here and then add just one or two to it.
"One of those is injured but we hope to complete a deal with him, one we hope will be completed in the next 48 hours and one will probably happen before the end of the transfer window.
"All in all, I'm happy with the group that's here."