Blues boss Andre Villas-Boas has decided to farm out 18-year-old McEachran, whose first-team opportunities at Stamford Bridge have been extremely limited this season.
The England Under-21 international has long been coveted by former Chelsea youth team boss Rodgers, who is confident of beating off reported competition from Bolton and Aston Villa to land the midfielder for the remainder of the campaign.
He told talkSPORT: "Hopefully, we can get that agreement confirmed in the next 24-48 hours. We've obviously made an enquiry.
"Josh is a boy I know well so we'll see if we can organise the move over the next few days.
"He'll be brilliant for the Swansea supporters and they'll get to witness one of the best talents this country's got.
"He's a player who sees passes from the sky. He's an incredible passer of the ball and I know the England set-up have very high hopes for him as well.
"He's a kid who can create chances and has wonderful vision on the ball - he fits perfectly into our team.
"I've got a really talented group of players at this level and he'll come into the squad and add to that for the second part of the season."