The 18-year-old has been available for a temporary move by his parent club Chelsea in order to gain some regular appearances.
Bolton Wanderers and Aston Villa were both said to be interested in signing the midfielder, but it now appears that Rodgers has captured the youngster until the end of the season.
"Hopefully, we can get that agreement confirmed in the next 24-48 hours," he said. "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."
McEachran has made 11 league appearances for the Blues.