The 26-year-old still needs to pass a medical but the Potters are moving closer to bolstering their midfield options.
As a USA international, he will also need to be granted a work permit before the move can be made official.
Edu would be the second American signed by Stoke this summer following the arrival of Geoff Cameron from Houston Dynamo.
He will also join a former Rangers team-mate at the Britannia Stadium in Jamie Ness.
Rangers have seen an exodus of talent this summer after being demoted to the fourth tier of Scottish football following their liquidation last season.
Prior to his move to the SPL in 2008, Edu made a name for himself winning MLS Rookie of the Year at Toronto FC.